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Question Number 130082 by benjo_mathlover last updated on 22/Jan/21

Given sin^(−1) (sin a+sin b)+sin^(−1) (sin a−sin b)=(π/2) what is the value of sin^2 a +sin^2 b.

Givensin−1(sina+sinb)+sin−1(sina−sinb)=π2whatisthevalueofsin2a+sin2b.

Answered by liberty last updated on 22/Jan/21

let sin^(−1) (sin a+sin b)=p ⇔ sin a + sin b = sin p let sin^(−1) (sin a−sin b)=q ⇔ sin a−sin b = sin q and p+q = (π/2); p=(π/2)−q ⇒ sin p = cos q ⇒sin^2 p = 1−sin^2 q or sin^2 p+sin^2 q=1 ⇒sin^2 a+2sin a sin b+sin^2 b =sin^2 p...(1) ⇒sin^2 a−2sin a sin b+sin^2 b=sin^2 q...(2) adding (1) and (2)give 2(sin^2 a+sin^2 b)= 1 then we get sin^2 a+sin^2 b = (1/2)

letsin−1(sina+sinb)=p⇔sina+sinb=sinpletsin−1(sina−sinb)=q⇔sina−sinb=sinqandp+q=π2;p=π2−q⇒sinp=cosq⇒sin2p=1−sin2qorsin2p+sin2q=1⇒sin2a+2sinasinb+sin2b=sin2p...(1)⇒sin2a−2sinasinb+sin2b=sin2q...(2)adding(1)and(2)give2(sin2a+sin2b)=1thenwegetsin2a+sin2b=12

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