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प्रश्न
Prove that
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`
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उत्तर
To prove
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`
Taking LHS, we get:
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]`
let `x=cos 2theta`
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]=tan^(-1) [(sqrt(1+cos2theta)-sqrt(1-cos2theta))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]`
`=tan^(-1)[(costheta-sintheta)/(costheta+sintheta)]`
`=tan^(-1)[(1-tantheta)/(1+tantheta)]`
`=tan^(-1) tan(pi/4-theta)`
`=(pi/4-theta)`
`=π/4−θ`
`=π/4−1/2cos^(−1) x`
`=RHS `
Hence proved.
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