3. 2sin3an 12453. 2sintan-1

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Solution

We have to prove that 2 sin −1 3 5 = tan −1 24 7 .

Consider sin −1 3 5 =x, then,

sinx= 3 5 cosx= 1− ( 3 5 ) 2 = 25−9 25 = 4 5

Another trigonometric function is,

tanx= sinx cosx = 3 5 4 5 = 3 4 x= tan −1 3 4

Substitute sin −1 3 5 = tan −1 3 4 to the left hand side of the given equation,

2 sin −1 3 5 =2 tan −1 3 4 = tan −1 ( 2( 3 4 ) 1− ( 3 4 ) 2 ) = tan −1 ( 3 2 7 16 ) = tan −1 ( 24 7 )

Hence, it is proved that 2 sin −1 3 5 = tan −1 24 7 .

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