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Find the value of "cos"(2cos^(-1)x+sin^(...

Find the value of `"cos"(2cos^(-1)x+sin^(-1)x)` at `x=1/5,` where `0lt=pi` and `-pi/2lt=sin^(-1)xlt=pi/2dot`

A

1

B

3

C

0

D

`-(2sqrt(6))/(5)`

Text Solution

Verified by Experts

The correct Answer is:
D
`cos(2cos^(-1)x+sin^(-1)x)`
`=cos[2(cos^(-1)x+sin^(-1)x)-sin^(-1)x]`
`=cos(pi-sin^(-1)x)=-cos(sin^(-1)x)" " [becausecos^(-1)x+sin^(-1)x=(pi)/(2)]`
`=-cos[sin^(-1)(-(1)/(5))] " " [becausex=(1)/(5)]`
`=-cos(-"cos"^(-1)(2sqrt(6))/(5))=-(2sqrt(6))/(5) " " [becausecos(-theta)=costheta] " " [becausesin^(-1)x=cos^(-1)sqrt(1-x^(2))]`
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