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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 6024 = 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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