Question

Number of solutions in [-1, 1], of the equation $si{n}^{-1}\frac{2}{3}x+si{n}^{-1}\frac{\sqrt{5}}{3}x=si{n}^{-1}x$ is:

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

The correct option is C 3

Trivially x = 0 satisfies the equation
$si{n}^{-1}\frac{2x}{3}+si{n}^{-1}\frac{x\sqrt{5}}{3}=si{n}^{-1}x...\left(1\right)$
For $x\ne 0$ on taking sine of both sides, we get
$\frac{2x}{3}\sqrt{1-\frac{5x^2}{9}}+\frac{x\sqrt{5}}{3}\sqrt{1-\frac{4x^2}{9}}=x\Rightarrow 2\sqrt{9-5x^2}+\sqrt{5}\sqrt{9-4x^2}=9[\because x\ne 0]\Rightarrow 4(9-5x^2)+81-36\sqrt{9-5x^2}=5(9-4x^2)\Rightarrow 72=36\sqrt{9-5x^2}\Rightarrow 4=9-5x^2\Rightarrow 5x^2=5\Rightarrow x=\pm 1$
On verification, we can easily show that x = 1 and - 1 both satisfy the equation (1).