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Prove that cos^(-1) {sqrt((1 + x)/(2))} ...

Prove that `cos^(-1) {sqrt((1 + x)/(2))} = (cos^(-1) x)/(2) , -1 lt x lt 1`

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Let `x = cos theta`, where `theta in [0, pi]`. Then,
`cos^(-1) {sqrt((1 + x)/(2)) = cos^(-1) {sqrt((1 + cos theta)/(2))}`
`= cos^(-1) {sqrt((2 cos^(2).(theta)/(2))/(2))}`
`= cos^(-1) (cos.(theta)/(2)) ( :' cos.(theta)/(2) gt 0)`
`= (theta)/(2) = (cos^(-1) x)/(2)`
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CENGAGE PUBLICATION-INVERSE TRIGONOMETRIC FUNCTIONS-All Questions
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  2. If x<0, then prove that cos^(-1)x=pi-sin^(-1)sqrt(1-x^2)

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  3. Prove that cos^(-1) {sqrt((1 + x)/(2))} = (cos^(-1) x)/(2) , -1 lt x l...

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  4. Prove that tan^(-1){x/(a+sqrt(a^2-x^2))}=1/2sin^(-1)(x/a) ,-a lt x lt ...

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  6. Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2tan^(-1)x^n ,0 lt x llt oo

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