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Question

If 0 < x <(1/√2) and ( sin -1 x/α)=( cos -1 x/β), then a value of sin ((2 π α/α+β)) is (A) 4 √(1-x2)(1-2 x2) (B) 4 x √(1-x2)(1-2 x2) (C) 2 x √(1-x2)(1-4 x2) (D) 4 √(1-x2)(1-4 x2)

Tardigrade Admin · Accepted Answer

( sin -1 x/α)=( cos -1 x/β)= k sin -1 x=k α cos -1 x= k β k =(π/2(α+β)) .....(i) sin ((2 π α/α+β))= sin (4 sin -1 x) =2 sin (2 sin -1 x) cos (2 sin -1 x) =4 x √1-x2(1-2 x2)

Q. If $0 < x < \frac{1}{2}$ and $\frac{s i n ^{- 1} x}{α} = \frac{c o s ^{- 1} x}{β}$ , then a value of $sin (\frac{2 π α}{α + β})$ is

$4 (1 - x^{2}) (1 - 2 x^{2})$

$4 x (1 - x^{2}) (1 - 2 x^{2})$

$2 x (1 - x^{2}) (1 - 4 x^{2})$

$4 (1 - x^{2}) (1 - 4 x^{2})$

$\frac{s i n ^{- 1} x}{α} = \frac{c o s ^{- 1} x}{β} = k$
$sin^{- 1} x = k α$
$cos^{- 1} x = k β$
$k = \frac{π}{2 ( α + β )} .....$ (i)
$sin (\frac{2 π α}{α + β}) = sin (4 sin^{- 1} x)$
$= 2 sin (2 sin^{- 1} x) cos (2 sin^{- 1} x)$
$= 4 x 1 - x^{2} (1 - 2 x^{2})$

Q. If 0<x<2​1​ and αsin−1x​=βcos−1x​, then a value of sin(α+β2πα​) is

4(1−x2)​(1−2x2)

4x(1−x2)​(1−2x2)

2x(1−x2)​(1−4x2)

4(1−x2)​(1−4x2)