Question

If cos^–1x > sin^–1x, then ______.

विकल्प

• `1/sqrt(2) < x ≤ 1`

• `0 ≤ x < 1/2`

• `-1 ≤ x  < 1/2`

• x > 0

Answer 1

If cos^–1x > sin^–1x, then `-1 ≤ x  < 1/2`.

Explanation:

Here, given that cos^–1x > sin^–1x

⇒ `sin[cos^-1x] > x`

⇒ `sin[sin^-1 sqrt(1 - x^2)] > x`

⇒ `sqrt(1 - x^2) > x`

⇒ `x < sqrt(1 - x^2)`

⇒ `x^2 < 1 - x^2`

⇒ `2x^2 < 1`

⇒ `x^2 < 1/2`

⇒ `x < +- 1/sqrt(2)`

We know that – 1 ≤ x ≤ 1

So – 1 ≤ x < `1/sqrt(2)`.