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Question

# 3 an 36+ sin17524. Sin

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Solution

## We have to prove that sin −1 8 17 + sin −1 3 5 = tan −1 77 36 . Consider sin −1 8 17 =x, then, sinx= 8 17 cosx= 1− ( 8 17 ) 2 = 225 289 = 15 17 Another trigonometric function is, tanx= sinx cosx = 8 15 x= tan −1 8 15 sin −1 8 17 = tan −1 8 15 Consider sin −1 3 5 =y, then, siny= 3 5 cosy= 1− ( 3 5 ) 2 = 16 25 = 4 5 Another trigonometric function is, tany= siny cosy = 3 4 y= tan −1 3 4 sin −1 3 5 = tan −1 3 4 Substitute sin −1 8 17 = tan −1 8 15 and sin −1 3 5 = tan −1 3 4 to the left hand side of the given equation, sin −1 8 17 + sin −1 3 5 = tan −1 8 15 + tan −1 3 4 = tan −1 ( 8 15 + 3 4 1− 8 15 × 3 4 ) = tan −1 32+45 60−24 = tan −1 77 36 Hence, it is proved that sin −1 8 17 + sin −1 3 5 = tan −1 77 36 .

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