If P = sin<sup>20</sup> θ + cos<sup>48</sup> θ, then the inequality that holds for all values of θ is
\(0 \le \sin^2\theta \le 1\) and \(0 \le \cos^2\theta \le 1\). Raising a number in [0, 1] to a higher power doesn't make it bigger, so
\(\sin^{20}\theta \le \sin^2\theta\) and \(\cos^{48}\theta \le \cos^2\theta\).
Therefore \(P \le \sin^2\theta + \cos^2\theta = 1\).
Also \(P > 0\), because \(\sin\theta\) and \(\cos\theta\) can't both be 0.
So \(0 < P \le 1\). (It equals 1 at \(\theta = 0\) or \(\frac{\pi}{2}\).)
Answer: (B)
Reinforce your conceptual grasp with related questions from NIMCET 2015:
Take timed full-length mock tests with authentic question distributions, negative marking (+12/-3 in Math), and instant All India Rank calculation.
Start Free Mock Test Series →