Question

Solve the equation : 7D%20%20%5C%3A%20cos%283x%29" id="TexFormula1" title=" \cos(x) - \sin(x) = \sqrt{2 } \: cos(3x)" alt=" \cos(x) - \sin(x) = \sqrt{2 } \: cos(3x)" align="absmiddle" class="latex-formula">
​

Answer 1

Answer:

General solution is

 $x = n \pi + \frac{\pi }{8}$

Step-by-step explanation:

Step(i):-

Given  cos x - sin x = √2 cos (3 x)

Dividing '√2' on both sides , we get

$\frac{1}{\sqrt{2} } cos (x) - \frac{1}{\sqrt{2} } sin (x) = \frac{\sqrt{2} cos (3 x)}{\sqrt{2} }$

we will use trigonometry formulas

a) Cos ( A + B) = Cos A Cos B - sin A sin B

b)  $cos \frac{\pi }{4} = \frac{1}{\sqrt{2} }$

Step(ii):-

$\frac{1}{\sqrt{2} } cos (x) - \frac{1}{\sqrt{2} } sin (x) = \frac{\sqrt{2} cos (3 x)}{\sqrt{2} }$

$cos (\frac{\pi }{4} ) cos x - sin(\frac{\pi }{4} ) sin x = cos 3x$

$cos (\frac{\pi }{4}+x ) = cos 3 x$

Step(iii):-

General solution of  cos x = cos ∝  is  x = 2 nπ+∝

we have $cos (\frac{\pi }{4}+x ) = cos 3 x$

The general solution of  $cos (\frac{\pi }{4}+x ) = cos 3 x$ is

⇒  $3 x = 2 n \pi + (\frac{\pi }{4}+x )$

⇒ $3 x- x = 2 n \pi + \frac{\pi }{4}$

 $2x = 2 n \pi + \frac{\pi }{4}$

final answer:-

General solution is

 $x = n \pi + \frac{\pi }{8}$