Solve the following for x-sin-11-x-2sin-1x-2

Sin^ 1(1 x) 2sin^ 1x=π/2 ... (i) ⇒sin^ 1(1 x)=π/2+2sin^ 1x ⇒ 1 x=sin(π/2+2sin^ 1x )=cos (2sin^ 1x) ⇒ 1 x=cos 2α where sin^ 1x=α i.e., x=sinα ⇒ 1 x=1 2sin^2 ...

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Standard XII

Mathematics

Theorems on Integration

Solve the fol...

Question

Solve the following for $x$ :

${sin}^{- 1} (1 - x) - 2 {sin}^{- 1} x = \frac{π}{2}$

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Solution

${sin}^{- 1} (1 - x) - 2 {sin}^{- 1} x = \frac{π}{2}$ ... (i)

$\Rightarrow {sin}^{- 1} (1 - x) = \frac{π}{2} + 2 {sin}^{- 1} x$

$\Rightarrow 1 - x = sin (\frac{π}{2} + 2 {sin}^{- 1} x) = cos (2 {sin}^{- 1} x)$

$\Rightarrow 1 - x = cos 2 α$ where ${sin}^{- 1} x = α$ i.e., $x = sin α$

$\Rightarrow 1 - x = 1 - 2 {sin}^{2} α$

$\Rightarrow 1 - x = 1 - 2 x^{2}$

$\Rightarrow 2 x^{2} - x = 0$

$\Rightarrow x (2 x - 1) = 0$

$\Rightarrow x = 0, \frac{1}{2}$

Since, $x = \frac{1}{2}$ does not satisfy equation (i)

$∴ x = 0$

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Solve the following for x: sin−1(1−x)−2sin−1x=π2

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Solve the following for $x$ :

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