Prove that 2 ^ -1 - + x 2 = ^ -1 ( + x + x ) - Vidyadip

Question

Prove that $2 \tan ^{-1}\left\{\sqrt{\frac{\alpha-\beta}{\alpha+\beta}} \tan \frac{x}{2}\right\}=\cos ^{-1}\left(\frac{\beta+\alpha \cos x}{\alpha+\beta \cos x}\right)$

✓

Answer

Suppose $\tan ^{-1}\left\{\sqrt{\frac{\alpha-\beta}{\alpha+\beta}} \tan \frac{x}{2}\right\}=\theta$
$\therefore \tan \theta =\sqrt{\frac{\alpha-\beta}{\alpha+\beta}} \tan \cdot \frac{x}{2}$
$\text { now L.H.S. } =2 \theta=\cos ^{-1}(\cos 2 \theta)$
$ =\cos ^{-1}\left(\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\right)\left[\because \cos 2 \theta=\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\right]$
Putting the values :
$=\cos ^{-1}\left(\frac{1-\frac{\alpha-\beta}{\alpha+\beta} \tan ^2 \frac{x}{2}}{1+\frac{\alpha-\beta}{\alpha+\beta} \tan ^2 \frac{x}{2}}\right)$
$=\cos ^{-1}\left(\frac{(\alpha+\beta) \cos ^2 \frac{x}{2}-(\alpha-\beta) \sin ^2 \frac{x}{2}}{(\alpha+\beta) \cos ^2 \frac{x}{2}+(\alpha-\beta) \sin ^2 \frac{x}{2}}\right)$
$=\cos ^{-1}\left(\frac{\beta\left(\cos ^2 \frac{x}{2}+\sin ^2 \frac{x}{2}\right)+\alpha\left(\cos ^2 \frac{x}{2}-\sin ^2 \frac{x}{2}\right)}{\alpha\left(\cos ^2 \frac{x}{2}+\sin ^2 \frac{x}{2}\right)+\beta\left(\cos ^2 \frac{x}{2}-\sin ^2 \frac{x}{2}\right)}\right)$
$=\cos ^{-1}\left(\frac{\beta+\alpha \cos x}{\alpha+\beta \cos x}\right)=\text { R.H.S. }$

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 inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}1 & 6 \\ -3 & 5 \end{bmatrix}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x e }^\text{x}\log\text{x}+\text{e}^\text{x}}{\text{x}\cos\text{y}}$

Find the length and the foot of the perpendicular from the point $(1, 1, 2)$ to the plane $\vec{\text{r}}.\big(\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}\big)+5=0.$

Find the equation of the plane passing through the point (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2y + 4z = 10.

If $\text{A}=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ the$n$ prove that $A^2 - A + 2I = O$.

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\text{dx}}{\text{a}\cos\text{x}+\text{b}\sin\text{x}}\text{ a},\text{b}>0$

Suppose a girl throws a die. If she gets $1$ or $2,$ she tosses a coin three times and notes the number of tails. If she gets $3, 4, 5$ or $6,$ she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', then what is the probability that she threw $3, 4, 5$ or $6$ with the die?

If function $f: R \rightarrow R , f(x)=x^2+2$ and $g: R \rightarrow R$ $g(x)=\frac{x}{x-1}, x \neq 1$ then find $f o g$ and $g o f$ and also find $( fog )(2)$ and $( gof )(-3)$ ?

Solve the following differential equation : $\left(x^2+1\right) \frac{d y}{d x}+2 x y=\sqrt{x^2+4}$

Find the direction cosines of the line $\frac{4-\text{x}}{2}=\frac{\text{y}}{6}=\frac{1-\text{z}}{3}.$ Also, reduce it to vector form