Question 1

Which of the following expressions is equivalent to $\frac { 5 \cot ^ { 2 } t } { \csc t }$ 
 (t $\neq$  $\pi$n, where n is a whole number)

Accepted Answer

A)   $\frac { 5 - 5 \sin ^ { 2 } t } { \sin t }$ 
B)  $\frac { - 1 - \sin ^ { 2 } t } { 5 \sin t }$ 
C)  $\frac { 5 - 5 \cos ^ { 2 } t } { \cos t }$ 
D)  $\frac { 5 + 5 \sin ^ { 2 } t } { \sin t }$ 
E)  $\frac { 5 + 5 \cos ^ { 2 } t } { \cos t }$ 
A)   $\frac { 5 - 5 \sin ^ { 2 } t } { \sin t }$ 
B)  $\frac { - 1 - \sin ^ { 2 } t } { 5 \sin t }$ 
C)  $\frac { 5 - 5 \cos ^ { 2 } t } { \cos t }$ 
D)  $\frac { 5 + 5 \sin ^ { 2 } t } { \sin t }$ 
E)  $\frac { 5 + 5 \cos ^ { 2 } t } { \cos t }$

Question 2

Use the cofunction identities to evaluate the expression without using a calculator. ​
Cos265° + cos225°
​

Accepted Answer

A) 3 
B) 4 
C) -3 
D) -1 
E) 1 
A) 3 
B) 4 
C) -3 
D) -1 
E) 1

Question 3

Rewrite $\ln | \sin \theta | - \ln | \cos \theta |$ as a single logarithm and then simplify the result.

Accepted Answer

A)   $\ln | \cos \theta |$ 
B)   $\ln | \csc \theta |$ 
C)   $\ln | \sec \theta |$ 
D)   $\ln | \tan \theta |$ 
E)   $\ln | \sin \theta |$ 
A)   $\ln | \cos \theta |$ 
B)   $\ln | \csc \theta |$ 
C)   $\ln | \sec \theta |$ 
D)   $\ln | \tan \theta |$ 
E)   $\ln | \sin \theta |$

Question 4

Using the grid provided, sketch the graph of the given function in the interval (-5, 5) and then determine the x-intercepts, if any. $$y = \sin \left( \frac { \pi x } { 2 } \right) - 1$$

Accepted Answer

A) (-3,0) and (1,0) 
B) (-2,0) 
C) (-2,0) and (2,0) 
D) no x-intercepts 
E) (-4,0), (0,0), and (4,0) 
A) (-3,0) and (1,0) 
B) (-2,0) 
C) (-2,0) and (2,0) 
D) no x-intercepts 
E) (-4,0), (0,0), and (4,0)

Question 5

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.  
  $\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$

Accepted Answer

A)   $\csc ^ { 2 } \alpha$ 
B)   $1 + \cot ^ { 2 } \alpha$ 
C)   $\sec ^ { 2 } \alpha$ 
D)   $1 / \cos ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$ 
E)   $\frac { 1 } { 1 - \cos ^ { 2 } \alpha }$ 
A)   $\csc ^ { 2 } \alpha$ 
B)   $1 + \cot ^ { 2 } \alpha$ 
C)   $\sec ^ { 2 } \alpha$ 
D)   $1 / \cos ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$ 
E)   $\frac { 1 } { 1 - \cos ^ { 2 } \alpha }$

Question 6

Solve the following equation. 
 $2 \sin x - 1 = 0$

Accepted Answer

A)   $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 4 \pi } { 3 } + 2 n \pi$ , where n is an integer 
B)   $x = \frac { \pi } { 6 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 6 } + 2 n \pi$ , where n is an integer 
C)   $x = \frac { \pi } { 3 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 3 } + 2 n \pi$ , where n is an integer 
D)   $x = \frac { \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is an integer 
E)   $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 4 \pi } { 3 } + 2 n \pi$ , where n is an integer. 
A)   $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 4 \pi } { 3 } + 2 n \pi$ , where n is an integer 
B)   $x = \frac { \pi } { 6 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 6 } + 2 n \pi$ , where n is an integer 
C)   $x = \frac { \pi } { 3 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 3 } + 2 n \pi$ , where n is an integer 
D)   $x = \frac { \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is an integer 
E)   $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 4 \pi } { 3 } + 2 n \pi$ , where n is an integer.

Question 7

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta $, where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 11 - x ^ { 2 } } , x = \sqrt { 11 } \sin \theta$

Accepted Answer

A)  $- \sqrt { 11 } \cos \theta$ 
B)  $\sqrt { 11 } \cos \theta$ 
C)  $\sqrt { 11 } \sin \theta$ 
D)  $11 \cos \theta$ 
E)  $11 \sin \theta$ 
A)  $- \sqrt { 11 } \cos \theta$ 
B)  $\sqrt { 11 } \cos \theta$ 
C)  $\sqrt { 11 } \sin \theta$ 
D)  $11 \cos \theta$ 
E)  $11 \sin \theta$

Question 8

Simplify the given expression algebraically. 
 $\cos ( x - \pi )$

Accepted Answer

A)  $\sin x$ 
B)  $\cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$ 
E) 1 
A)  $\sin x$ 
B)  $\cos x$ 
C)  $- \sin x$ 
D)  $- \cos x$ 
E) 1

Question 9

Convert the expression.   $\tan \frac { a } { 2 }$

Accepted Answer

A)   $\csc a - \tan a$ 
B)  $\csc a + \cot a$ 
C)   $\cos a - \cot a$ 
D)  $\cos a + \cot a$ 
E)  $\csc a - \cot a$ 
A)   $\csc a - \tan a$ 
B)  $\csc a + \cot a$ 
C)   $\cos a - \cot a$ 
D)  $\cos a + \cot a$ 
E)  $\csc a - \cot a$

Question 10

Use the product-to-sum formula to write the given product as a sum or difference.   $12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$

Accepted Answer

A)   $6 - 6 \cos \frac { \pi } { 12 }$ 
B)  $6 \sin \frac { \pi } { 6 } + 6 \cos \frac { \pi } { 6 }$ 
C)  $6 \sin \frac { \pi } { 3 } + 6 \sin 0$ 
D)  $6 + 6 \cos \frac { \pi } { 12 }$ 
E)  $- 6 \sin \frac { \pi } { 12 }$ 
A)   $6 - 6 \cos \frac { \pi } { 12 }$ 
B)  $6 \sin \frac { \pi } { 6 } + 6 \cos \frac { \pi } { 6 }$ 
C)  $6 \sin \frac { \pi } { 3 } + 6 \sin 0$ 
D)  $6 + 6 \cos \frac { \pi } { 12 }$ 
E)  $- 6 \sin \frac { \pi } { 12 }$

Question 11

Use the figure below to determine the exact value of the given function.   $\csc 2 \theta$

Accepted Answer

A)   $\csc 2 \theta = \frac { 13 } { 5 }$ 
B)   $\csc 2 \theta = \frac { 13 } { 7 }$ 
C)   $\csc 2 \theta = \frac { 13 } { 5 }$. 
D)   $\csc 2 \theta = \frac { 13 } { 9 }$ 
E)   $\csc 2 \theta = \frac { 13 } { 12 }$ 
A)   $\csc 2 \theta = \frac { 13 } { 5 }$ 
B)   $\csc 2 \theta = \frac { 13 } { 7 }$ 
C)   $\csc 2 \theta = \frac { 13 } { 5 }$. 
D)   $\csc 2 \theta = \frac { 13 } { 9 }$ 
E)   $\csc 2 \theta = \frac { 13 } { 12 }$

Question 12

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval [0, 2$\pi $).   $2 \cos ^ { 2 } x - \cos x - 1 = 0$

Accepted Answer

A)   $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $x = 0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
C)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $x = \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
E)  solution does not exist 
A)   $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $x = 0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
C)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $x = \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$ 
E)  solution does not exist

Question 13

Use the given values to evaluate (if possible) two trigonometric functions $\tan \varphi$ and $\csc \psi$ . 
 $\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }$

Accepted Answer

A)   $\begin{array} { l } 
\tan \varphi = - \frac { 1 } { 5 } \
\csc \psi = \sqrt { 10 }
\end{array}$ 
B)   $\begin{array} { l } 
\tan \varphi = \frac { 1 } { 5 } \
\csc \psi = \sqrt { 10 }
\end{array}$ 
C)   $\begin{array} { l } 
\tan \varphi = \frac { 1 } { 5 } \
\csc \psi = - \sqrt { 10 }
\end{array}$ 
D)   $\begin{array} { l } 
\tan \varphi = - 5 \
\csc \psi = \sqrt { 10 }
\end{array}$ 
E)   $\begin{array} { l } 
\tan \varphi = - \frac { 1 } { 5 } \
\csc \psi = - \sqrt { 10 }
\end{array}$ 
A)   $\begin{array} { l } 
\tan \varphi = - \frac { 1 } { 5 } \
\csc \psi = \sqrt { 10 }
\end{array}$ 
B)   $\begin{array} { l } 
\tan \varphi = \frac { 1 } { 5 } \
\csc \psi = \sqrt { 10 }
\end{array}$ 
C)   $\begin{array} { l } 
\tan \varphi = \frac { 1 } { 5 } \
\csc \psi = - \sqrt { 10 }
\end{array}$ 
D)   $\begin{array} { l } 
\tan \varphi = - 5 \
\csc \psi = \sqrt { 10 }
\end{array}$ 
E)   $\begin{array} { l } 
\tan \varphi = - \frac { 1 } { 5 } \
\csc \psi = - \sqrt { 10 }
\end{array}$

Question 14

Use a double-angle formula to rewrite the expression. ​
10 cos2 x - 5
​

Accepted Answer

A) 5 cos x 
B) ​cos 5x 
C) ​10 cos 2x 
D) ​10 cos x 
E) ​5 cos 2x 
A) 5 cos x 
B) ​cos 5x 
C) ​10 cos 2x 
D) ​10 cos x 
E) ​5 cos 2x

Question 15

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\cos ^ { 2 } 20 ^ { \circ } + \cos ^ { 2 } 57 ^ { \circ } + \cos ^ { 2 } 70 ^ { \circ } + \cos ^ { 2 } 33 ^ { \circ }$

Accepted Answer

A) 1 
B) 2 
C) -1 
D) 0 
E)  $\frac { 1 } { 2 }$ 
A) 1 
B) 2 
C) -1 
D) 0 
E)  $\frac { 1 } { 2 }$

Question 16

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

Accepted Answer

A)  $\tan ( 7 x )$ 
B)  $\tan ( 3 x )$ 
C)  $\tan ( 5 x )$ 
D)  $\tan ( - x )$ 
E)  $\tan ( - 5 x )$ 
A)  $\tan ( 7 x )$ 
B)  $\tan ( 3 x )$ 
C)  $\tan ( 5 x )$ 
D)  $\tan ( - x )$ 
E)  $\tan ( - 5 x )$

Question 17

Solve the following equation.  $\cos ^ { 2 } x + \cos x = 0$

Accepted Answer

A)   $x = \frac { 3 \pi } { 4 } + n \pi \text { and } x = \frac { 5 \pi } { 4 } + n \pi$ , where n is an integer 
B)   $x = \pi + 2 n \pi \text { and } x = \frac { \pi } { 2 } + n \pi$ , where n is an integer 
C)   $x = \frac { 3 \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is an integer 
D)   $x = n \pi \text { and } x = \frac { 3 \pi } { 2 } + n \pi$ , where n is an integer 
E)   $x = \pi + n \pi \text { and } x = \frac { \pi } { 2 } + 2 n \pi$ , where n is an integer 
A)   $x = \frac { 3 \pi } { 4 } + n \pi \text { and } x = \frac { 5 \pi } { 4 } + n \pi$ , where n is an integer 
B)   $x = \pi + 2 n \pi \text { and } x = \frac { \pi } { 2 } + n \pi$ , where n is an integer 
C)   $x = \frac { 3 \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is an integer 
D)   $x = n \pi \text { and } x = \frac { 3 \pi } { 2 } + n \pi$ , where n is an integer 
E)   $x = \pi + n \pi \text { and } x = \frac { \pi } { 2 } + 2 n \pi$ , where n is an integer

Question 18

Simplify the expression algebraically. $5 \sin \left( \frac { \pi } { 6 } + x \right)$

Accepted Answer

A)  $\frac { 5 } { 2 }$ $( \cos x - \sqrt { 3 } \sin x )$ 
B)  $\frac { 5 } { 2 }$ $( \cos x + \sqrt { 3 } \sin x )$ 
C)  $\frac { 5 } { 2 }$ $( \sin x - \sqrt { 3 } \cos x )$ 
D)  $\frac { 5 } { 2 }$ $( \cos x + \sin x )$ 
E)  $\frac { 5 } { 2 }$ $( \sin x + \sqrt { 3 } \cos x )$ 
A)  $\frac { 5 } { 2 }$ $( \cos x - \sqrt { 3 } \sin x )$ 
B)  $\frac { 5 } { 2 }$ $( \cos x + \sqrt { 3 } \sin x )$ 
C)  $\frac { 5 } { 2 }$ $( \sin x - \sqrt { 3 } \cos x )$ 
D)  $\frac { 5 } { 2 }$ $( \cos x + \sin x )$ 
E)  $\frac { 5 } { 2 }$ $( \sin x + \sqrt { 3 } \cos x )$

Question 19

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II.)

Accepted Answer

A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E)  $\cos ( u + v ) = \frac { 56 } { 65 }$ 
A)  $\cos ( u + v ) = \frac { 28 } { 65 }$ 
B)  $\cos ( u + v ) = - \frac { 16 } { 65 }$ 
C)  $\cos ( u + v ) = \frac { 33 } { 65 }$ 
D)  $\cos ( u + v ) = - \frac { 28 } { 65 }$ 
E)  $\cos ( u + v ) = \frac { 56 } { 65 }$

Question 20

Evaluate the following expression.(x $\neq$$\pi$n, y $\neq$  $\pi$/12 + $\pi$n, where n is a whole number) 
 $\frac { 7 \tan 6 x + 7 \cot 6 y } { \tan 6 x \cot 6 y }$

Accepted Answer

A) 7 tan 6x + 7 cot 6y 
B) 7 tan 6y - 7 cot 6x 
C) 7 tan 6y + 7 cot 6x 
D) - (7 tan 6y + 7 cot 6x) 
E) 7 tan 6x - 7 cot 6y 
A) 7 tan 6x + 7 cot 6y 
B) 7 tan 6y - 7 cot 6x 
C) 7 tan 6y + 7 cot 6x 
D) - (7 tan 6y + 7 cot 6x) 
E) 7 tan 6x - 7 cot 6y

Which of the following expressions is equivalent to $\frac { 5 \cot ^ { 2 } t } { \csc t }$ (t $\neq$ $\pi$ n, where n is a whole number)

Use the cofunction identities to evaluate the expression without using a calculator. Cos²65° + cos²25°

Rewrite $\ln | \sin \theta | - \ln | \cos \theta |$ as a single logarithm and then simplify the result.

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$

Solve the following equation. $2 \sin x - 1 = 0$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 11 - x ^ { 2 } } , x = \sqrt { 11 } \sin \theta$

Simplify the given expression algebraically. $\cos ( x - \pi )$

Convert the expression. $\tan \frac { a } { 2 }$

Use the product-to-sum formula to write the given product as a sum or difference. $12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$

Use the figure below to determine the exact value of the given function. $\csc 2 \theta$ $Use the figure below to determine the exact value of the given function. \csc 2 \theta$

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval [0, 2 $\pi$ ). $2 \cos ^ { 2 } x - \cos x - 1 = 0$

Use the given values to evaluate (if possible) two trigonometric functions $\tan \varphi$ and $\csc \psi$ . $\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }$

Use a double-angle formula to rewrite the expression. 10 cos²x - 5

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\cos ^ { 2 } 20 ^ { \circ } + \cos ^ { 2 } 57 ^ { \circ } + \cos ^ { 2 } 70 ^ { \circ } + \cos ^ { 2 } 33 ^ { \circ }$

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

Solve the following equation. $\cos ^ { 2 } x + \cos x = 0$

Simplify the expression algebraically. $5 \sin \left( \frac { \pi } { 6 } + x \right)$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II.)

Evaluate the following expression.(x $\neq$ $\pi$ n, y $\neq$ $\pi$ /12 + $\pi$ n, where n is a whole number) $\frac { 7 \tan 6 x + 7 \cot 6 y } { \tan 6 x \cot 6 y }$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 5: Analytic Trigonometry

Which of the following expressions is equivalent to 5cot⁡2tcsc⁡t\frac { 5 \cot ^ { 2 } t } { \csc t }csct5cot2t​ (t ≠\neq= π\piπ n, where n is a whole number)

Use the cofunction identities to evaluate the expression without using a calculator. ​ Cos265° + cos225° ​

Rewrite ln⁡∣sin⁡θ∣−ln⁡∣cos⁡θ∣\ln | \sin \theta | - \ln | \cos \theta |ln∣sinθ∣−ln∣cosθ∣ as a single logarithm and then simplify the result.

Using the grid provided, sketch the graph of the given function in the interval (-5, 5) and then determine the x-intercepts, if any. y=sin⁡(πx2)−1y = \sin \left( \frac { \pi x } { 2 } \right) - 1y=sin(2πx​)−1

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. cot⁡2α+cot⁡2α⋅tan⁡2α\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alphacot2α+cot2α⋅tan2α

Solve the following equation. 2sin⁡x−1=02 \sin x - 1 = 02sinx−1=0

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ\theta θ , where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ . 11−x2,x=11sin⁡θ\sqrt { 11 - x ^ { 2 } } , x = \sqrt { 11 } \sin \theta11−x2​,x=11​sinθ

Simplify the given expression algebraically. cos⁡(x−π)\cos ( x - \pi )cos(x−π)

Convert the expression. tan⁡a2\tan \frac { a } { 2 }tan2a​

Use the product-to-sum formula to write the given product as a sum or difference. 12sin⁡π6cos⁡π612 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }12sin6π​cos6π​

Use the figure below to determine the exact value of the given function. csc⁡2θ\csc 2 \thetacsc2θ

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval [0, 2 π\pi π ). 2cos⁡2x−cos⁡x−1=02 \cos ^ { 2 } x - \cos x - 1 = 02cos2x−cosx−1=0

Use the given values to evaluate (if possible) two trigonometric functions tan⁡φ\tan \varphitanφ and csc⁡ψ\csc \psicscψ . cot⁡φ=−5,sin⁡ψ=1010\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }cotφ=−5,sinψ=1010​​

Use a double-angle formula to rewrite the expression. ​ 10 cos2 x - 5 ​

Write the given expression as the tangent of an angle. tan⁡3x+tan⁡4x1−tan⁡3xtan⁡4x\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }1−tan3xtan4xtan3x+tan4x​

Solve the following equation. cos⁡2x+cos⁡x=0\cos ^ { 2 } x + \cos x = 0cos2x+cosx=0

Simplify the expression algebraically. 5sin⁡(π6+x)5 \sin \left( \frac { \pi } { 6 } + x \right)5sin(6π​+x)

Find the exact value of cos⁡(u+v)\cos ( u + v )cos(u+v) given that sin⁡u=513\sin u = \frac { 5 } { 13 }sinu=135​ and cos⁡v=−45\cos v = - \frac { 4 } { 5 }cosv=−54​ .(Both u and v are in Quadrant II.)

Evaluate the following expression.(x ≠\neq=π\piπ n, y ≠\neq= π\piπ /12 + π\piπ n, where n is a whole number) 7tan⁡6x+7cot⁡6ytan⁡6xcot⁡6y\frac { 7 \tan 6 x + 7 \cot 6 y } { \tan 6 x \cot 6 y }tan6xcot6y7tan6x+7cot6y​

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Which of the following expressions is equivalent to $\frac { 5 \cot ^ { 2 } t } { \csc t }$ (t $\neq$ $\pi$ n, where n is a whole number)

Use the cofunction identities to evaluate the expression without using a calculator. Cos²65° + cos²25°

Rewrite $\ln | \sin \theta | - \ln | \cos \theta |$ as a single logarithm and then simplify the result.

Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$

Solve the following equation. $2 \sin x - 1 = 0$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 11 - x ^ { 2 } } , x = \sqrt { 11 } \sin \theta$

Simplify the given expression algebraically. $\cos ( x - \pi )$

Convert the expression. $\tan \frac { a } { 2 }$

Use the product-to-sum formula to write the given product as a sum or difference. $12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$

Use the figure below to determine the exact value of the given function. $\csc 2 \theta$ $Use the figure below to determine the exact value of the given function. \csc 2 \theta$

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval [0, 2 $\pi$ ). $2 \cos ^ { 2 } x - \cos x - 1 = 0$

Use the given values to evaluate (if possible) two trigonometric functions $\tan \varphi$ and $\csc \psi$ . $\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }$

Use a double-angle formula to rewrite the expression. 10 cos²x - 5

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

Solve the following equation. $\cos ^ { 2 } x + \cos x = 0$

Simplify the expression algebraically. $5 \sin \left( \frac { \pi } { 6 } + x \right)$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II.)

Evaluate the following expression.(x $\neq$ $\pi$ n, y $\neq$ $\pi$ /12 + $\pi$ n, where n is a whole number) $\frac { 7 \tan 6 x + 7 \cot 6 y } { \tan 6 x \cot 6 y }$