Question

Number of principal solution(s) of the $\sqrt{sin x}-\frac{1}{\sqrt{sin ⁡ x}}=cos⁡x$ equation is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$\sqrt{sin x}-\frac{1}{\sqrt{sin ⁡ x}}=cos⁡x$
$sin x+\frac{1}{sin ⁡ x}-2=\left(cos\right)^{2}x=\left(1 - \left(sin\right)^{2} x\right) \, $
Let $sin x=t$
$t+\frac{1}{t}-2=1-t^{2}$
$t^{2}-2t+1=t\left(1 - t\right)\left(1 + t\right)$
$\Rightarrow \, \left(1 - t\right)\left[\left(1 - t\right) - t \left(1 + t\right)\right]=0$
$\Rightarrow \left(1 - t\right)\left[1 - 2 t - t^{2}\right] \, =0$
$t=1,t^{2}+2t-1=0$
$sin x=1,sin⁡x=\frac{- 2 \pm \sqrt{4 + 4}}{2}$
$sin x=-1\pm\sqrt{2}$
$sin x=1,\Rightarrow x=\frac{\pi }{2}$
also $sin x=\sqrt{2}-1$ and $cos x < 0$
$\Rightarrow $ one solution is $\left(0 ,2 \pi \right)$