Question

tan inverse 1/2 = π/4 - 1/2cos inverse 4/5

Answer 1

36.86989764584401


is the answer

Answer 2

Answer and Explanation:

To prove : $\tan^{-1}(\frac{1}{2})=\frac{\pi}{4}-\frac{1}{2}\cos^{-1}(\frac{4}{5})$

Proof :

Let $\cos^{-1}(\frac{4}{5})=x$

$\cos x=\frac{4}{5}$

Applying identity,

$\frac{1-\tan^2(\frac{x}{2})}{1+\tan^2(\frac{x}{2})}=\frac{4}{5}$

$5-5\tan^2(\frac{x}{2})=4+4\tan^2(\frac{x}{2})$

$\tan^2(\frac{x}{2})=\frac{1}{9}$

$\tan(\frac{x}{2})=\frac{1}{3}$

Now, Take the expression as

$\frac{1}{2}=\tan(\frac{\pi}{4}-\frac{1}{2}x)$

Take RHS,

$=\tan(\frac{\pi}{4}-\frac{x}{2})$

Apply formula, $\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\times \tan B}$

$=\frac{\tan (\frac{\pi}{4})-\tan (\frac{x}{2})}{1+\tan(\frac{\pi}{4})\times \tan (\frac{x}{2})}$

$=\frac{1-\tan (\frac{x}{2})}{1+\tan (\frac{x}{2})}$

Substitute, $\tan(\frac{x}{2})=\frac{1}{3}$

$=\frac{1-\frac{1}{3}}{1+\frac{1}{3}}$

$=\frac{3-1}{3+1}$

$=\frac{2}{4}$

$=\frac{1}{2}$

=LHS

Hence proved.