First slide

Compound angles

Question

If $A = \cos (\cos x) + \sin (\cos x) .$ Then least and greatest value of A are:

Moderate

A

$0 a n d 2$
B

$- 1 a n d 1$
C

$- \sqrt{2} a n d \sqrt{2}$
D

none of the above

Solution

$\begin{array}{l} W e h a v e A = \cos (\cos x) + \sin (\cos x) \\ = \sqrt{2} \{\cos (\cos x) \cos \frac{π}{4} + \sin (\cos x) \sin \frac{π}{4}\} \\ = \sqrt{2} \{\cos (\cos x - \frac{π}{4})\} \\ Since - 1 < \cos (\cos x - \frac{π}{4}) \leq 1 \\ - \sqrt{2} < \sqrt{2} \cos (\cos x - \frac{π}{4}) \leq \sqrt{2} \\ \Rightarrow - \sqrt{2} < A \leq \sqrt{2} \end{array}$

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