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Sin-1(1-x)-2sin-1x=π2, tan 'x' is equal to -

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Question

`sin^-1(1 - x) - 2sin^-1 x = pi/2`, tan 'x' is equal to

Options

  • `0, 1/2`

  • `1, 1/2`

  • 0

  • `1/2`

MCQ

Solution

0

Explanation:

`sin^-1(1 - x) - 2sin^-1 x = pi/2`

⇒ `- 2 sin x = pi/2 - sin^-1 = 2pi(4 - x)`

⇒ `- 2sin^-1x = cos^-1(1 - x)`

Let `sin^-1x` = θ

⇒ sin θ = x

⇒ θ = `cos^-1 (sqrt(1 - x^2))`

∴ `sin^-1x = cos^-1 (sqrt(1 - x^2))`

From equation (1) we have

`- 2cos^-1 (sqrt(1 - x^2)) = cos^-1(1 - x)`

Put x = sin y

`- 2cos^-1 (sqrt(1 - sin^2y)) = cos^1(1 - sin y)`

⇒ `- 2cos^-1 (cos y) = cos^-1 (1 - sin y)`

⇒ `- 2y = cos^-1(1 - sin y)`

⇒ `1 - sin y = cos (- 2y) = cos 2y`

⇒ `1 - sin y = 1 - sin^2y`

⇒ `2sin^2y - sin y` = 0

⇒ `sin y (2 sin y - 1)` = 0

⇒ `sin y` = 0 or `1/2`

∴ x = 0 or x = `1/2`

But when x = `1/2`, does not satisfy the equation, Thus x = 0 is the only solution.

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