The number of real values of \(x \in[0,2 \pi]-\left\{\frac{\pi}{2}, \frac{3 \pi}{2}\right\}\) satisfying the…
The number of real values of \(x \in[0,2 \pi]-\left\{\frac{\pi}{2}, \frac{3 \pi}{2}\right\}\) satisfying the equation \(|\cos x|^{2 \sin ^2 x-3 \sin x+1}=1\), is
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Solution
Given that,
\(\begin{gathered}
|\cos x|^{2 \sin ^2 x-3 \sin x+1}=1 \\
\therefore \quad|\cos x|=1 \text { or } 2 \sin ^2 x-3 \sin x+1=0 \\
\cos x= \pm 1 \text { or }(2 \sin x-1)(\sin x-1)=0 \\
x=0, \pi, 2 \pi \text { or } \sin x=\frac{1}{2} \text { or } \sin x=1 \\
x=0, \pi, 2 \pi \text { or } x=\frac{\pi}{6}, \frac{5 \pi}{6} \text { or } x=\frac{\pi}{2} \\
\Rightarrow x=0, \pi, 2 \pi, \frac{\pi}{6}, \frac{5 \pi}{6}\left[\because x \in[0,2 \pi]-\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\right]
\end{gathered}\)
\(\therefore\) The number of real values of \(x\) is 5.