Solve cos-1 x-2 cos-1-1-x2-

cos ^ 1 x=2cos ^ 1√(1+x2) let x=cos 2θ from L.H.S cos ^ 1 #160; x=cos ^ 2 #160; θ #160; x=cos ^ 2 #160; θ #160; #16 ...

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Use of Trigonometric Table

Solve cos-1 ...

Question

Solve
${cos}^{- 1} x = 2 {cos}^{- 1} \sqrt{\frac{1 + x}{2}}$ .

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Solution

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c o s^{- 1} x = 2 s i n^{- 1} \sqrt{\frac{1 - x}{2}} = 2 c o s^{- 1} \sqrt{\frac{1 + x}{2}}

s i n^{- 1} (- x) = - s i n^{- 1} x, c o s^{- 1} (- x) = π - c o s^{- 1} x (- 1 \leq x \leq 1)

{cos}^{- 1} (2 x \sqrt{1 - x^{2}}) = (\frac{π}{2}) - 2 {cos}^{- 1} x, \frac{1}{\sqrt{2}} \leq x \leq 1

x \in [- π, 3 π]

cos \frac{(\sqrt{2} + 1) x}{2} cos \frac{(\sqrt{2} - 1) x}{2} = 1

c o s^{- 1} x

2 c o s^{- 1} x = s i n^{- 1} (2 x \sqrt{1 - x^{2}})

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