Question

If tan²ø=1+2tan²alpha
Prove that,
Sin²ø=1/2(1+sin²alpha)​

Answer 1

Given :

$\mapsto\sf \tan ^2\phi=1+2\tan^{2} \alpha$

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Proof :

$\:\:\:\:\:\:\:\:\:\sf\tan^2=1+2\tan^2\alpha$

$\sf\implies \sec^2\phi-1=1+2\tan^2\alpha$

$\sf\implies\cos^2\phi=\frac{1}{2(1+\tan^2\alpha)}$

$\sf\implies 1-\sin^2\phi=\frac{1}{2(1+\tan^2\alpha)}$

$\sf\implies\sin^2\phi=\frac{1}{2}\{2-\frac{1}{(\sec^2\alpha)}\}$

$\sf\implies\sin^2\phi=\frac{1}{2}\{1+1-\cos^2\alpha\}$

$\sf\:\:\:\:\:\:\therefore{\underline{\boxed{\sf{\sin^2\phi=\frac{1}{2}\{1+\sin^2\alpha\}}}}$

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Formula Required :

$\sf 1) \sin^2\theta+\cos^2\theta=1$

$\sf 2) \sec^2\theta-\tan^2\theta=1$

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HOPE THIS IS HELPFUL...