Question

If sin | sin-s-cos-1 x 1 then find the value of xsin+osx-1, then find the value of x14.

Answer 1

It is given that sin( sin −1 1 5 + cos −1 x )=1.

On simplifying we get,

sin( sin −1 1 5 + cos −1 x )=1 sin( sin −1 1 5 )cos( cos −1 x )+cos( sin −1 1 5 )sin( cos −1 x )=1 1 5 ×x+cos( sin −1 1 5 )sin( cos −1 x )=1 x 5 +cos( sin −1 1 5 )sin( cos −1 x )=1 (1)

Assume, sin −1 1 5 =y, then,

siny= 1 5 cosy= 1− ( 1 5 ) 2 cosy= 2 6 5 y= cos −1 ( 2 6 5 )

This gives sin −1 1 5 = cos −1 ( 2 6 5 ).

Also, assume cos −1 x=z, then,

cosz=x sinz= 1− x 2 z= sin −1 ( 1− x 2 )

This gives cos −1 x= sin −1 ( 1− x 2 ).

Substitute sin −1 1 5 = cos −1 ( 2 6 5 ) and cos −1 x= sin −1 ( 1− x 2 ) in equation (1),

x 5 +cos( cos −1 ( 2 6 5 ) )sin( sin −1 ( 1− x 2 ) )=1 x 5 + 2 6 5 × 1− x 2 =1 x+2 6 × 1− x 2 =5 2 6 × 1− x 2 =5−x

Square both sides of the above equation,

4×6×( 1− x 2 )=25+ x 2 −10x 24−24 x 2 =25+ x 2 −10x 25 x 2 −10x+1=0 ( 5x−1 ) 2 =0

Further simplify,

5x−1=0 5x=1 x= 1 5

Therefore, the value of x is 1 5 .