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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
Question
State True or False for the following statement:
If $\tan(\pi\cos\theta)=\cot(\pi\sin\theta),$ then $\cos\Big(\theta-\frac{\pi}{4}\Big)=\pm\frac{1}{2\sqrt{2}}$
✓

Answer

True.
Solution:
Given that, $\tan(\pi\cos\theta)=\cot(\pi\sin\theta)$
$\Rightarrow\tan(\pi\cos\theta)=\tan\Big(\frac{\pi}{2}-\pi\sin\theta\Big)$
$\Rightarrow\pi\cos\theta=\frac{\pi}{2}-\pi\sin\theta$
$\Rightarrow\pi\cos\theta+\pi\sin\theta=\frac{\pi}{2}$
$\Rightarrow\cos\theta+\sin\theta=\frac{1}{2}$
$\Rightarrow\frac{1}{\sqrt2}\cos\theta+\frac{1}{\sqrt2}\sin\theta=\frac{1}{2\sqrt2}$
$\Rightarrow\cos\frac{\pi}{4}\cos\theta+\sin\frac{\pi}{4}\sin\theta=\frac{1}{2\sqrt2}$$$
$\Rightarrow\cos\Big(\theta-\frac{\pi}{4}\Big)=\pm\frac{1}{2\sqrt2}$$\Big[\because\cos\Big(\theta-\frac{\pi}{4}\Big)$or $\cos\Big(\frac{\pi}{4}-\theta\Big)\Big]$
Hence, the given statement is 'true'.

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