Composition of Inverse Trigonometric Functions and Trigonometric Functions

Q. The value of $co{s}^{-1}\left(cos10\right)$=

1. $10$
2. $4\pi -10$
3. $2\pi +10$
4. $2\pi -10$

Q. the value of cot-1(cot(-10))

Q. Let $f_1\left(x\right)={\sin }^{-1}\left(\cos \right({\sin }^{2}x\left)\right),$ $f_2\left(x\right)={\cos }^{-1}\left(\sin \right({\cos }^{2}x\left)\right),$ $f_3\left(x\right)={\sin }^{-1}\left(\cos \right({\cos }^{2}x\left)\right)$ and $f_4\left(x\right)={\cos }^{-1}\left(\sin \right({\sin }^{2}x\left)\right).$ Then which of the following is (are) CORRECT?

1. $\sum _{i=1}^4{f}_i\left(x\right)=2\pi -2$
2. $\sum _{i=1}^4(-1{)}^i{f}_i(x)=0$
3. $\sum _{i=1}^4(-1{)}^{\left[i/3\right]}{f}_i(x)=0,$ where $[.]$ denotes the greatest integer function
4. $f_1\left(x\right)>f_2\left(x\right)\Rightarrow \cos 2x>0$

Q. For any positive integer $n$, define $f_n:(0,\infty )\to R$ as
$f_n\left(x\right)=\sum _{j=1}^n{\tan }^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right)$ for all $x\in (0,\infty )$
$($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $(\frac{-\pi }{2},\frac{\pi }{2})$ $)$
Then, which of the following statement(s) is (are) TRUE?

1. $\sum _{j=1}^5{\tan }^2\left(f_j\right(0\left)\right)=55$
2. $\sum _{j=1}^{10}(1+f_j^{\prime }(0\left)\right){\sec }^2\left(f_j\right(0\left)\right)=10$
3. For any fixed positive integer $n,\underset{x\to \infty }{lim}\tan \left(f_n\right(x\left)\right)=\frac{1}{n}$
4. For any fixed positive integer $n,\underset{x\to \infty }{lim}{\sec }^2\left(f_n\right(x\left)\right)=1$

Q. 178. The range of f(theta) = 3cos²theta - 8 root 3 cos theta.sin theta + 5sin²theta - 7 is given by
i) [-7 , 7]
ii)[-10 , 4]
iii)[-4 , 4]
iv)[-10 , 7]

Q. The number of non-integral values of $x$ satisfying $\sin \left[2{\cos }^{-1}\right\{\cot \left(2{\tan }^{-1}x\right)\left\}\right]=0$ is

Q. If $si{n}^{-1}\left(sin\frac{33\pi }{7}\right)+co{s}^{-1}\left(cos\frac{46\pi }{7}\right)+ta{n}^{-1}(-tan\frac{13\pi }{8})+co{t}^{-1}\left(cot(-\frac{19\pi }{8})\right)=\frac{a\pi }{b}$ Where a and b are in their lowest form, then (a + b) equal to

1. 17
2. 20
3. 23
4. None of these

Q.

$ta{n}^{-1}\frac{1}{\sqrt{x^2-1}}=$

1. $\frac{x}{2}+cose{c}^{-1}x$
2. $\frac{x}{2}+se{c}^{-1}x$
3. $cose{c}^{-1}x$
4. $se{c}^{-1}x$

Q. Which of the following is true?

1. $tan|ta{n}^{-1}x|=|x|$
2. $cot|co{t}^{-1}x|=|x|$
3. $ta{n}^{-1}|tanx|=|x|$
4. $sin|si{n}^{-1}x|=|x|$