The sum of possible values of $x$ for $\tan ^ { - 1 } ( x + 1 ) + \cot ^ { - 1 } \left( \frac { 1 } { x - 1 } \right) = \tan ^ { - 1 } \left( \frac { 8 } { 31 } \right)$ is:
(1) $- \frac { 32 } { 4 }$
(2) $- \frac { 31 } { 4 }$
(3) $- \frac { 30 } { 4 }$
(4) $- \frac { 33 } { 4 }$

If the inverse trigonometric functions take principal values, then $\cos ^ { - 1 } \left( \frac { 3 } { 10 } \cos \left( \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) \right) + \frac { 2 } { 5 } \sin \left( \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) \right) \right)$ is equal to
(1) 0
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 6 }$

The set of all values of $k$ for which $\left( \tan ^ { - 1 } x \right) ^ { 3 } + \left( \cot ^ { - 1 } x \right) ^ { 3 } = \mathrm { k } \pi ^ { 3 } , x \in R$, is the interval
(1) $\left[ \frac { 1 } { 32 } , \frac { 7 } { 8 } \right)$
(2) $\left( \frac { 1 } { 24 } , \frac { 13 } { 16 } \right)$
(3) $\left[ \frac { 1 } { 48 } , \frac { 13 } { 16 } \right]$
(4) $\left[ \frac { 1 } { 32 } , \frac { 9 } { 8 } \right)$

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}x - 2\sin^{-1}x = \cos^{-1}(2x)$ is equal to
(1) 0
(2) 1
(3) $\frac{1}{2}$
(4) $-\frac{1}{2}$

The value of $\tan^{-1}\left(\frac{\cos\frac{15\pi}{4} - 1}{\sin\frac{\pi}{4}}\right)$ is equal to
(1) $-\frac{\pi}{4}$
(2) $-\frac{\pi}{8}$
(3) $-\frac{5\pi}{12}$
(4) $-\frac{4\pi}{9}$

$\tan ^ { - 1 } \frac { 1 + \sqrt { 3 } } { 3 + \sqrt { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 + 4 \sqrt { 3 } } { 6 + 3 \sqrt { 3 } } } =$
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 6 }$

If the value of $\frac { 3 \cos 36 ^ { \circ } + 5 \sin 18 ^ { \circ } } { 5 \cos 36 ^ { \circ } - 3 \sin 18 ^ { \circ } }$ is $\frac { a \sqrt { 5 } - b } { c }$, where $a , b , c$ are natural numbers and $\operatorname { gcd } ( a , c ) = 1$, then $a + b + c$ is equal to : (1) 40 (2) 52 (3) 50 (4) 54

For $\alpha , \beta , \gamma \neq 0$. If $\sin ^ { - 1 } \alpha + \sin ^ { - 1 } \beta + \sin ^ { - 1 } \gamma = \pi$ and $(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3 \alpha \beta$, then $\gamma$ equals
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(4) $\sqrt { 3 }$

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16 \left( \left( \sec ^ { - 1 } x \right) ^ { 2 } + \left( \operatorname { cosec } ^ { - 1 } x \right) ^ { 2 } \right)$ is:
(1) $24 \pi ^ { 2 }$
(2) $22 \pi ^ { 2 }$
(3) $31 \pi ^ { 2 }$
(4) $18 \pi ^ { 2 }$

If $\frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 4 }$, then $\cos ^ { - 1 } \left( \frac { 12 } { 13 } \cos x + \frac { 5 } { 13 } \sin x \right)$ is equal to
(1) $x - \tan ^ { - 1 } \frac { 4 } { 3 }$
(2) $x + \tan ^ { - 1 } \frac { 4 } { 5 }$
(3) $x - \tan ^ { - 1 } \frac { 5 } { 12 }$
(4) $x + \tan ^ { - 1 } \frac { 5 } { 12 }$

If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1}\left\{ \beta + \frac{(1 + \beta^{2})}{(\alpha - \beta)} \right\} + \cot^{-1}\left\{ \gamma + \frac{(1 + \gamma^{2})}{(\beta - \gamma)} \right\} + \cot^{-1}\left\{ \alpha + \frac{(1 + \alpha^{2})}{(\gamma - \alpha)} \right\}$ is equal to:
(1) $\pi$
(2) 0
(3) $\frac{\pi}{2} - (\alpha + \beta + \gamma)$
(4) $3\pi$

Let $\mathrm{S} = \{x: \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1)\}$. Then $\sum_{x \in \mathrm{S}}(2x-1)^2$ is equal to \_\_\_\_ .

Given that $0 < x < \frac { \pi } { 2 }$ and $\cot \mathrm { x } - 3 \tan \mathrm { x } = \frac { 1 } { \sin 2 \mathrm { x } }$, what is $\sin ^ { 2 } x$?
A) $\frac { 1 } { 9 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 7 }$

Given that $0 < x < \frac { \pi } { 2 }$, $$\frac { \sec ( x ) - 1 } { 2 } = \frac { 3 } { \sec ( x ) + 1 }$$ the equality holds.\ Accordingly, what is the value of $\tan ( x )$?\ A) $\sqrt { 2 }$\ B) $\sqrt { 3 }$\ C) $\sqrt { 5 }$\ D) $\sqrt { 6 }$\ E) $\sqrt { 7 }$

Let $0 < \mathrm { x } < \frac { \pi } { 2 }$. $\sec x \cdot \tan x \cdot ( 1 - \sin x ) = \frac { 1 } { 4 }$ Accordingly, what is the value of $\csc x$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) 2
E) 3

$$\frac { \cos ^ { 2 } \left( 80 ^ { \circ } \right) + 5 \sin ^ { 2 } \left( 80 ^ { \circ } \right) - 3 } { \cos \left( 50 ^ { \circ } \right) }$$
Which of the following is this expression equal to?
A) $\cot \left( 50 ^ { \circ } \right)$
B) $\sec \left( 20 ^ { \circ } \right)$
C) $\sec \left( 40 ^ { \circ } \right)$
D) $\operatorname { cosec } \left( 20 ^ { \circ } \right)$
E) $\operatorname { cosec } \left( 40 ^ { \circ } \right)$

In the figure, using the points $\mathrm { P } ( 0,1 )$ and $\mathrm { S } ( 1,0 )$ on the unit circle with center O and the positive directed angle $\theta$ that the line segment RO makes with the x-axis, new trigonometric functions are defined as follows:
$$\begin{aligned} & \text { kas } \theta = | \mathrm { RS } | \\ & \text { sas } \theta = | \mathrm { RP } | \end{aligned}$$
Accordingly,
$$\frac { \mathrm { kas } ^ { 2 } \theta } { 2 - \operatorname { sas } ^ { 2 } \theta }$$
For $\theta$ values where this expression is defined, which of the following is it equal to?
A) $\sin ( 2 \theta )$
B) $\cos ^ { 2 } ( 2\theta )$
C) $\sec \theta$
D) $\tan \left( \frac { \theta } { 2 } \right)$

The simplified form of the expression
$$\frac{1 - \cos(4x)}{\sin(4x) + 2 \cdot \sin(2x)}$$
is which of the following?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\sec x$ E) $\operatorname{cosec} x$

Let $0 < x < \frac{\pi}{2}$. Given that
$$2 \cdot \cos^{2} x + 9 \cdot \sin^{2} x + 2 \cdot \sin(2x) = 9$$
what is the value of $\cot x$?
A) $\frac{4}{7}$ B) $\frac{7}{6}$ C) $\frac{3}{5}$ D) $\frac{2}{3}$ E) $\frac{5}{2}$