Explanation:

(P)

\( \sin^{6} x = 1 - \cos^{4} x \)

\( \sin^{6} x = (1 - \cos^{2} x)(1 + \cos^{2} x) \) ; \( \sin^{6} x = \sin^{2} x (1 + \cos^{2} x) \)

\( \sin^{2} x = 0 \) or \( \sin^{4} x = 1 + \cos^{2} x \)

\( (\sin^{4} x)_{\max} = 1 \)

\( (1 + \cos^{2} x)_{\min} = 1 \)

\( \sin x = \pm 1 \) or \( \sin x = 0 \)

Number of element in \( [ -\pi, \pi ] = 5 \)

(Q)

\( \sin^{2} x + \cos^{6} x = 1 \)

\( \cos^{6} x = \cos^{2} x \) ; \( \cos^{2} x = 0 \) or \( 1 \) ; \( \cos x = 0, 1 \) or \( -1 \).

Number of element in \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] = 3 \)

(R)

\( \cos^{2}\!\left(\frac{x}{2}\right) - \sin^{2} x = \frac{1}{2} \)

\( \sin^{2} x = 4 \sin^{2}\!\left(\frac{x}{2}\right) \cos^{2}\!\left(\frac{x}{2}\right) \)

Let \( \cos^{2}\!\left(\frac{x}{2}\right) = t \)

\( t - 4 t(1 - t) = \frac{1}{2} \)

\( 2t - 8t + 8t^{2} = 1 \)

\( 8t^{2} - 6t - 1 = 0 \)

\( t = \frac{6 \pm \sqrt{36 + 32}}{16} \) ; \( t = \frac{3 \pm \sqrt{17}}{8} \) ; \( \cos^{2}\!\left(\frac{x}{2}\right) = \frac{3 + \sqrt{17}}{8} \)

Number of element in \( [ -\pi, \pi ] = 2 \)

(S)

\( 6 \sin^{2}\!\left(\frac{x}{2}\right) - \cos 3x = 3 \)

\( - \cos 3x = 3 \left( 1 - 2 \sin^{2}\!\left(\frac{x}{2}\right) \right) \)

\( 3 \cos 3x - 4 \cos^{3} x = 3 \cos x \)

\( -4 \cos^{3} x = 0 \)

\( \cos x = 0 \)

Number of elements in \( [ -2\pi, 2\pi ] = 4 \).

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