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Home » Prove that ????????????−????(√????+???? −√????−????)/(√????+???? + √????−????) =????/????-????/????????????????−???? ???? , −????√????≤ ???? ≤ ????[Hint: Put x = cos 2 θ]
Prove that ????????????−????(√????+???? −√????−????)/(√????+???? + √????−????) =????/????-????/????????????????−???? ???? , −????√????≤ ???? ≤ ????[Hint: Put x = cos 2 θ]
CBSE | Class 12 | Guides | Inverse Trigonometric Functions | Maths | NCERT
Prove that ????????????−????(√????+???? −√????−????)/(√????+???? + √????−????) =????/????-????/????????????????−???? ???? , −????√????≤ ???? ≤ ????[Hint: Put x = cos 2 θ]

Putting x=\cos 2 \theta \quad so that, \theta=\frac{1}{2} \cos ^{-1} x

LHS =\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)

=\tan ^{-1}\left(\frac{\sqrt{1+\cos 2 \theta}-\sqrt{1-\cos 2 \theta}}{\sqrt{1+\cos 2 \theta}+\sqrt{1-\cos 2 \theta}}\right)

=\tan ^{-1}\left(\frac{\sqrt{2 \cos ^{2} \theta}-\sqrt{2 \sin ^{2} \theta}}{\sqrt{2 \cos ^{2} \theta}+\sqrt{2 \sin ^{2} \theta}}\right)

=\tan ^{-1}\left(\frac{\sqrt{2} \cos \theta-\sqrt{2} \sin \theta}{\sqrt{2} \cos \theta+\sqrt{2} \sin \theta}\right)

Dividing every term by \sqrt{2} \cos \theta we get,

=\tan ^{-1}\left(\frac{1-\tan \theta}{1+\tan \theta}\right)

=\tan ^{-1}\left(\frac{\tan \frac{\pi}{4}-\tan \theta}{1+\tan \frac{\pi}{4} \tan \theta}\right)

=\tan ^{-1} \tan \left(\frac{\pi}{4}-\theta\right)

=\frac{\pi}{4}-\theta

=\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x

LHS =RHS

Hence proved

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