Prove that sin–1 (8 17 )+ ^ -1 (3 5 )= ^ -1 (36 85 ). - Vidyadip

Question

Prove that sin^–1 $\Bigg(\frac{8}{17}\Bigg)+\sin^{-1}\Bigg(\frac{3}{5}\Bigg)=\cos^{-1}\Bigg(\frac{36}{85}\Bigg).$

✓

Answer

Writing sin^-1$\Bigg(\frac{18}{17}\Bigg)=\tan^{-1}\frac{8}{15}$ and sin^-1$\Bigg(\frac{3}{5}\Bigg)=\tan^{-1}\frac{3}{4}$
$\therefore$ LHS = tan^-1$\frac{8}{15}+\tan^{-1}\frac{3}{4}=\tan^{-1}$$\Bigg(\frac{\frac{8}{15}+\frac{3}{4}}{1-\frac{8}{15}\cdot\frac{3}{4}}\Bigg)=\tan^{1}\Bigg(\frac{77}{36}\Bigg)$
Getting tan^–1$\Bigg(\frac{77}{36}\Bigg)=\cos^{-1}\Bigg(\frac{36}{85}\Bigg).$

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