# Inverse Trigonometric Functions - Class 12 - Mathematics

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## Extra Questions - Inverse Trigonometric Functions | NCERT | Mathematics | Class 12

Find the value of the expression $\tan(\frac{\sin^{-1} x + \cos^{-1}x}{2})$ when $x = \frac{\sqrt{3}}{2}$.

To start with, we need to simplify $\tan(\frac{\sin^{-1} x + \cos^{-1}x}{2})$. To do this, we'll utilize the relationship between the sine and cosine inverse functions.

$$ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} $$

This is a general identity because $\sin(\sin^{-1} x) = x$ and $\cos(\cos^{-1} x) = x$, and also $\cos(\sin^{-1} x) = \sqrt{1-x^2}$ and $\sin(\cos^{-1} x) = \sqrt{1-x^2}$. Adding $\sin^{-1}x$ and $\cos^{-1}x$ yields:

$$ \sin^{-1} x + \cos^{-1} x = \sin^{-1} x + \sin^{-1}(\cos(\sin^{-1} x)) = \sin^{-1} x + \sin^{-1}(\sqrt{1-x^2}) $$

Given $\sin A = x$ and $\cos A = \sqrt{1-x^2}$ for the same angle $A$, the addition of these inverses maps to $\pi/2$. This implies that $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$.

Now, substituting this into our original expression, we have:

$$ \tan\left(\frac{\sin^{-1} x + \cos^{-1} x}{2}\right) = \tan\left(\frac{\pi}{4}\right) $$

And we know that:

$$ \tan\left(\frac{\pi}{4}\right) = 1 $$

Thus, the value of $\tan(\frac{\sin^{-1} x + \cos^{-1} x}{2})$ when $x = \frac{\sqrt{3}}{2}$ is $1$.

The pair of lines represented by $3ax^2+5xy+(a^2-2)y^2=0$ are perpendicular to each other for

Option A) Two values of $a$

Option B) $\forall a$

Option C) For one value of $a$

Option D) For no value of $a$

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If $\text{3A} - \text{B} = \begin{bmatrix} 5 & 0 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}$, then find the matrix $A$

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Let $z = (\cos x)^5$ and $y = \sin x$. Then the value of $2 \frac{d^{2} z}{d y^{2}}$ at $x = \frac{2 \pi}{9}$ is

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The range of the function $f(x)=\frac{x+3}{|x+3|}$, $x \neq -3$, is

A) ${3,-3}$

B) $\mathrm{R}-{-3}$

C) all positive integers

D) ${-1,1}$

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If $f(x)=\log_e(1-x)$ and $g(x)=[x]$, then determine each of the following functions: (i) $f+g$ (ii) $fg$ (iii) $\frac{f}{g}$ (iv) $\frac{g}{f}$

Also, find $(f+g)(-1)$, $(fg)(0)$, $\left(\frac{f}{g}\right)\left(\frac{1}{2}\right)$, and $\left(\frac{g}{f}\right)\left(\frac{1}{2}\right)$.

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The number of solutions of the equation $[\sin x]=[1+\sin x]+[1-\cos x]$, where $[.]$ denotes the greatest integer function, is

A) 1 solution

B) 2 solutions

C) 3 solutions

D) No solution

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Find $\frac{dy_{i}}{dx}$ for the following expression: $$ y = \tan^{-1} \frac{3x - x^{3}}{1 - 3x^{2}} - \frac{1}{\sqrt{3}}, \text{where} \ -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}} .

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If $0<x, y<\pi$ and $\cos x + \cos y - \cos(x+y) = \frac{3}{2}$, then $\sin x + \cos y$ is equal to:

A) $\frac{1+\sqrt{3}}{2}$

B) $\frac{1-\sqrt{3}}{2}$

C) $\frac{\sqrt{3}}{2}$

D) $\frac{1}{2}`

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If $y = \sin\left(x^\circ\right)$, find $\frac{dy}{dx}$.

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If $x = \cos t \left(3 - 2 \cos^{2} t\right)$ and $y = \sin t \left(3 - 2 \sin^{2} t\right)$, find the value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$.

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The number of real solution(s) to the equation $\left(\sin^{-1} x\right)^{3} + \left(\cos^{-1} x\right)^{3} = 7\left(\tan^{-1} x + \cot^{-1} x\right)^{3}$ is:

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If $y = \sin t - \cos t$ and $x = \sin t + \cos t$, then $\frac{dy}{dx}$ at $t = \frac{\pi}{6}$ is:

A) 1

B) $2 + \sqrt{3}$

C) 0

D) $\frac{\sqrt{3} - 1}{2}$