Download App

Inverse Trigonometric Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSInverse Trigonometric Functions5 Marks
Question
Prove that $\cos \left[\tan ^{-1}\left\{\sin \left(\cot ^{-1} x\right)\right\}\right]=\sqrt{\frac{1+x^2}{2+x^2}}$
✓

Answer

$\text { L.H.S. }=\cos \left[\tan ^{-1} \sin \left(\cot ^{-1} x\right)\right]$
$\text { Suppose } \cot ^{-1} x=t \Rightarrow \cot t=x$
$\therefore \operatorname{cosec}^2 t=1+\cot ^2 t$
$\operatorname{cosec} t=\sqrt{1+x^2}$
$\therefore \sin t=\frac{1}{\sqrt{1+x^2}}$
$\therefore \text { L.H.S. }=\cos \left[\tan ^{-1} \sin t\right]$
$=\cos \left[\tan ^{-1} \frac{1}{\sqrt{1+x^2}}\right]$
$\operatorname{suppose} \tan ^{-1} \frac{1}{\sqrt{1+x^2}}=Z \Rightarrow \tan Z=\frac{1}{\sqrt{1+x^2}}$
$\therefore \sec ^2 Z=1+\tan ^2 Z$
$=1+\frac{1}{1+x^2}=\frac{1+x^2+1}{1+x^2}$
$=\frac{2+x^2}{1+x^2}$
$\therefore \sec Z=\sqrt{\frac{2+x^2}{1+x^2}} \Rightarrow \cos Z=\sqrt{\frac{1+x^2}{2+x^2}}$
$\text { So, } \text { L.H.S. }=\cos Z$
$=\sqrt{\frac{1+x^2}{2+x^2}}=\text { R.H.S. }$

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 "5 Marks Questions" from Inverse Trigonometric Functions→Inverse Trigonometric 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 equation of the plane through the intersection of the planes 3x - 4y + 5z = 10 and 2x + 2y - 3z = 4 and parallel to the line x = 2y = 3z.
→
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i}+2\hat{j}-\hat{k}\ \text{and}\ -\hat{i}+\hat{j}+\hat{k}$ respectively, in the ratio 2 : 1
  1. Internally.
  2. Externally.
→
From a lot of 30 bulbs that includes 6 defective bulbs, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
→
Find the distance between the parallel planes $2x - y + 3z − 4 = 0$ and $6x - 3y + 9z + 13 = 0.$
→
The two adjacent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}-5\hat{\text{k}}$ and $2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}.$ Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\cos\text{x}+\sin\text{x}}{\cos\text{x}-\sin\text{x}}\Big), \frac{\pi}{4}<\text{x}<\frac{\pi}{4}$
→
Evaluate the following integrals:
$\int(2\text{x}-5)\sqrt{2+3\text{x}-\text{x}^2}\text{dx}$
→
Prove that:
  1. $\text{P}(\text{A})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})$
  2. $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})+\text{P}(\text{A}\cap\bar{\text{B}})+\text{P}(\bar{\text{A}}\cap\text{B})$
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\sin^2\text{x}=\frac{1}{\text{x}\log\text{x}}$
→
A box of constant volume $c$ is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?                                                     
→