Question

Of sin^(-1)x + tan^(-1)x = pi/2 then prove that 2x^(2) + 1 = 5^(1/2) plzz solve this

Answer 1

this is the answer fr this question

Answer 2

Answer and explanation:

Given : $\sin^{-1}x+\tan^{-1}x=\frac{\pi}{2}$

To prove : $2x^{2} + 1 = 5^{\frac{1}{2}}$ ?

Solution :

We know,

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

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

$\tan^{-1}x=\cos^{-1}x$

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

$x=\frac{\sqrt{1-x^2}}{x}$

$x^2=\sqrt{1-x^2}$

Squaring both side,

$x^4=1-x^2$

$x^4+x^2-1=0$

Using quadratic formula,

$x^2=\frac{-1\pm\sqrt{1-4\cdot 1\cdot(-1)}}{2}$

$x^2=\frac{-1\pm\sqrt{5}}{2}$

$2x^2=-1\pm\sqrt{5}$

$2x^2+1=\sqrt{5}$

Hence proved.