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Q.
$sinax+cosax$ and $\left|sin x\right|+\left|cos x\right|$ are periodic with the same fundamental period, if $a$ equals to
NTA AbhyasNTA Abhyas 2022
Solution:
$sinax+cosax=\sqrt{2}sin\left(a x + \frac{\pi }{4}\right)$
So, fundamental period $=\frac{2 \pi }{\left|a\right|}$
Let, $T$ be the fundamental period of $\left|sin x\right|+\left|cos x\right|$
So $\left|sin \left(\right. x + T \left.\right)\right|+\left|cos \left(\right. x + T \left.\right)\right|= \, \left|sin x\right|+\left|cos x\right|.$
Squaring both sides, we get,
$1+\left|s i n \left(\right. 2 x + 2 T \left.\right)\right|=1+\left|sin 2 x\right|$
Therefore, the fundamental period of $\left|s i n x\right|+\left|c o s x\right|$ is $\frac{\pi }{2}$
Hence, $\frac{2 \pi }{\left|a\right|}=\frac{\pi }{2}$
$\Rightarrow a=\pm4$