Composition of Inverse Trigonometric Functions and Trigonometric Functions

Q. Find the value of ${\tan }^{-1}\left(\tan \left(-6\right)\right).$

1. $-6$
2. $\pi -6$
3. $-2\pi +6$
4. $2\pi -6$

Q.

Write the value of the tan inverse of $0.6666$ in degrees and minutes.

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. The value of $sin\left(co{t}^{-1}\right(cos\left(ta{n}^{-1}x\right)\left)\right)$ is equal to

1. $\sqrt{\frac{x^2+1}{x^2+2}}$
2. $\sqrt{\frac{x+2}{x^2+1}}$
3. $\sqrt{\frac{x^2-1}{x^2+2}}$
4. $\sqrt{\frac{x^2+2}{x^2+1}}$

Q.

The value of $\sinh \left(x+\mathit{2}\pi ni\right)$ is

1. $\frac{e^x+e^{-x}}{2}$

2. $\frac{e^x-e^{-x}}{2}$

3. $\frac{e^x-e^{-x}}{2i}$

4. $\frac{e^x+e^{-x}}{2i}$

Q. $\frac{1}{2}co{s}^{-1}\left(\frac{1-x}{1+x}\right)=$

1. $co{t}^{-1}\sqrt{x}$
2. $ta{n}^{-1}\sqrt{x}$
3. $ta{n}^{-1}x$
4. $co{t}^{-1}x$

Q. The set of values of p for which $x^2-px+si{n}^{-1}\left(sin4\right)>0$ for all real x is given by :

1. $(–4,4)$
2. $(-\infty ,-4)\cup (4,\infty )$
3. $\varphi$
4. None of these

Q.

$si{n}^{-1}[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}]=$

1. $si{n}^{-1}x-si{n}^{-1}\sqrt{1-x}$
   

2. $si{n}^{-1}x+si{n}^{-1}\sqrt{1-x}$
   

3. $si{n}^{-1}x-si{n}^{-1}\sqrt{x}$
   

4. None

Q. The value of $co{s}^{-1}\left(cos12\right)-si{n}^{-1}\left(sin14\right)$ is

1. $-2$
2. $8\pi -26$
3. None of these
4. $4\pi +2$

Q. The principal value of ${\cos }^{-1}(-\sin \frac{7\pi }{6})$

1. $\frac{7\pi }{3}$
2. $\frac{5\pi }{3}$
3. $\frac{\pi }{3}$
4. $-\frac{\pi }{3}$

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. If $x={\sin }^{-1}\left(\sin 10\right)$ and $y={\cos }^{-1}\left(\cos 10\right)$, then $y-x$ is equal to :

1. $0$
2. $\pi$
3. $10$
4. $7\pi$

Q. If $y\left(x\right)={\cot }^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right),x\in (\frac{\pi }{2},\pi ),$ then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is

1. $\frac{1}{2}$
2. $-1$
3. $-\frac{1}{2}$
4. $0$