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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`

Text Solution

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`cos(2cos^-1 x + sin^-1 x) `at `x=1/5`
`=> sin^-1 x + cos^-1 x = pi/2`
`(cos^-1x)= sqrt(1-x^2)`
`=> cos(2cos^-1 x + sin^-1x) = cos(cos^-1 x+ (cos^-1 x + sin^-1 x))`
`=> cos(pi/2 + cos^-1 x) = - sin(cos^-1 x )`
`= sin (cos^-1(1/5))`
`= - sqrt(1- (1/5)^2)`
`= sqrt(24/25) = -2sqrt6/5`
...
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