Prove that `sin^(-1)(8/(17))+sin^(-1)(3/5)=cos^(-1)((36)/(85))`

Explanation

Let `sin ^{-1}left(frac{8}{17})=alpha and sin ^{-1}left(frac{3}{5})=beta`
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Rightarrow sin alpha=frac{8}{17} and sin beta=frac{3}{5}`
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Rightarrow cos alpha=sqrt{1-sin ^{2} alpha} and cos beta=sqrt{1-sin ^{2} beta}`
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Rightarrow cos alpha=sqrt{1-frac{64}{289}} and cos beta=sqrt{1-frac{9}{25}}`
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Rightarrow cos alpha=sqrt{frac{289-64}{289}} and cos beta=sqrt{frac{25-9}{25}}`
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Rightarrow cos alpha=sqrt{frac{225}{289}} and cos beta=sqrt{frac{16}{25}}`
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Rightarrow cos alpha=frac{15}{17}`
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Rightarrow cos (alpha+beta)=cos alpha cdot cos beta-sin alpha cdot sin beta`
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