Prove the following results: 2 ^ -1 3 5 = ^ -1 24 7 - Vidyadip

Question

Prove the following results:
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

✓

Answer

$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$ $\text{L.H.S}=2\sin^{-1}\frac{3}{5}$ $=2\times\tan^{-1}\begin{bmatrix}\frac{\frac{3}{5}}{\sqrt{1-\Big(\frac{3}{5}\Big)^2}}\end{bmatrix}$ $\bigg\{\text{Since }\sin^{-1}\text{x}=\tan^{-1}\frac{\text{x}}{\sqrt{1-\text{x}^2}}\bigg\}$ $=2\tan^{-1}\Bigg(\frac{\frac{3}{5}}{\frac{4}{5}}\Bigg)$ $=2\tan^{-1}\Big(\frac{3}{4}\Big)$ $=\tan^{-1}\Bigg(\frac{2\times\frac{3}{4}}{1-\big(\frac{3}{4}\big)^2}\Bigg)$ $\Big\{\text{Since }2\tan^{-1}\text{x}=\tan^{-1}\frac{2\text{x}}{1-\text{x}^2}\Big\}$ $=\tan^{-1}\bigg(\frac{\frac{3}{2}}{\frac{7}{16}}\bigg)$ $=\tan^{-1}\Big(\frac{24}{7}\Big)$ $=\text{R.H.S}$So,
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from INVERSE TRIGNOMETRIC FUNCTIONS→INVERSE TRIGNOMETRIC FUNCTIONS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the local maxima and local minima of the function. Find also the local maximum and the local minimum value: f(x) = sinx - cosx, 0 < x < 2$ \pi$.

Let A = [-1, 1]. Then, discuss whether the following functions from A to itself are one-one, onto or bijective:
$\text{f(x)}=\frac{\text{x}}{2}$

Evaluate the following integrals:
$\int\cot^5\text{x}\text{ cosec}^4\text{x}\text{ dx}$

Find the value of $\int \frac{1}{\sqrt{\sin ^3 x \sin (x+\alpha)}} d x$.

Construct the composition table for $\times _6$ on set $S = \{0, 1, 2, 3, 4, 5\}.$

An ant is moving along the vector $\overrightarrow{l_1}=\hat{\imath}-2 \hat{\jmath}+3 \hat{k}$ Few sugar crystals are kept along the vector $\overrightarrow{l_2}=3 \hat{\imath}-2 \hat{j}+\hat{k}$ which is inclined at an angle $\theta$ with the vector $\overrightarrow{l_1}$. Then find the angle $\theta$. Also find the scalar projection of $\overrightarrow{l_1}$ on $\overrightarrow{l_2}$

Evaluate the following integrals:$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$

Find the area bounded by the curve $\text{y}=\sin\text{x}$ between x = 0 and $\text{x}=2\pi.$

If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.

Evaluate the integral : $\int \sqrt{\cot \theta} d \theta$