Question

The period of cos x sin (pi/4-x) is

Answer 1

Answer:

π

Step-by-step explanation:

py

Answer 2

Step-by-step explanation:

Given that: The period of cos x sin (pi/4-x) is?

Solution: To find the period of given function

$cos \: x \: sin \big( \frac{\pi}{4} - x\big)$

First open sin(π/4-x) with the formula sin(A-B)

$sin(A-B) = sin \: A \: cos \: B - cos \: A \: sin \: B \\ \\ sin \big( \frac{\pi}{4} - x \big) = sin \: \frac{\pi}{4} \: cos \: x - cos \: \frac{\pi}{4} \: sin \: x \\ \\ \because \: sin \: \frac{\pi}{4} = cos \: \frac{\pi}{4} = \frac{1}{ \sqrt{2} } \\ \\ sin\big( \frac{\pi}{4} - x \big)= \frac{1}{ \sqrt{2} } \: cos \: x - \frac{1}{ \sqrt{2} } \: sin \: x \\ \\ = \frac{1}{ \sqrt{2} } (cos \: x - sin \: x)$

Now,multiply cos x

$cos \: x \: sin \big( \frac{\pi}{4} - x\big)= \frac{1}{ \sqrt{2} } \Big( {cos}^{2}x - sin \: x \: cos \: x\Big)$

Multiply and divide by 2

$\frac{1}{ \sqrt{2} }\Bigg ( \frac{1}{2}\Big (2 {cos}^{2} x - 2sin \: x \: cos \: x\Big)\Bigg) \\ \\ \frac{1}{2 \sqrt{2} } (1 + cos \: 2x - sin \: 2x) \\ \\ cos \: x \: sin \big( \frac{\pi}{4} - x\big)= \frac{1}{2 \sqrt{2} } + \frac{cos \: 2x}{2 \sqrt{2} } - \frac{sin \: 2x}{2 \sqrt{2} }$

We know that period of cos x=2π

and sin x= 2π

Thus

Period of cos 2x= 2π/2= π

Period of sin 2x= 2π/2= π

Period of complete given function is LCM of both cos 2x and sin2x

Thus , LCM(π,π) = π

Period of cos x sin (pi/4-x) is π.

Hope it helps you.