Prove that :cos−1x=2sin−1(21−x).

R.E.F image Solution -cos−1x=2sin−121−xR.H.SLet y=2sin−121−x2y=cos−1(21+x)cos2(2y)=21+xcosy=2cos22y−1=1+x−1cosy=xy=cos−1x=LHS

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Prove that : { ^{ - 1}}x = 2{ ^{ - 1}}({{1-x}{2}} ).

Question

Prove that :
$cos^{- 1} x = 2 sin^{- 1} (\frac{1 - x}{2}) .$

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Prove that :

cos^{- 1} x = 2 sin^{- 1} (\frac{1 - x}{2}) .

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R.E.F image

Solution -

co s^{- 1} x = 2 s i n^{- 1} \frac{1 - x}{2}

R.H.S

Let

y = 2 s i n^{- 1} \frac{1 - x}{2}

\frac{y}{2} = co s^{- 1} (\frac{1 + x}{2})

co s^{2} (\frac{y}{2}) = \frac{1 + x}{2}

cos y = 2 co s^{2} \frac{y}{2} - 1 = 1 + x - 1

cos y = x

y = co s^{- 1} x

= L H S

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