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Which of the following inequalities are TRUE? (P) pi/3 > tan⁻¹(pi/3) (Q) pi/3 < tan⁻¹(pi/3) (R) pi/4 < cos⁻¹(pi/4) (S) pi/4 > cos⁻¹(pi/4)

1. P and R only
2. P and S only
3. Q and R only
4. Q and S only

Correct answer: P and R only

Solution

Since tan⁻¹(x) < x for all positive x, we have pi/3 > tan⁻¹(pi/3), making P true. Since pi/4 ~ 0.785 and cos(pi/4) ~ 0.707 < pi/4, and cos⁻¹ is a decreasing function, cos⁻¹(pi/4) > cos⁻¹(cos(pi/4)) = pi/4, making R true.

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