Question

-

Sin5x + sin3x =
cos5x + Cos3x​

Answer 1

Answer:

$\\\\\tt x = \dfrac{\pi}{4}$

Step-by-step explanation:

RHS:

$\\\tt cos \ 5x+ cos \ 3x = 2cos(\dfrac{5x+3x}{2} )cos(\dfrac{5x-3x}{2} )\\\\\tt = 2 cos (4x) cos(x)$

LHS:

$\\\tt -sin(5x) + sin (3x) = sin(3x)-sin(5x) \\\tt =2sin(\dfrac{5x-3x}{2} ) cos(\dfrac{5x+3x}{2} )\\\\\tt = 2sin (x) cos(4x)$

Equating both sides:

$\\\tt 2 cos(4x)cos(x) = 2 sin(x) cos(4x)\\\\\tt \implies cos(x) = sin(x) \\\tt \implies tan(x) = 1 \\\\\tt \implies x = \dfrac{\pi}{4}$

Formulas used:

• $\\\tt cos \ \alpha+ cos \ \beta = 2cos(\dfrac{\alpha + \beta}{2} )cos(\dfrac{\alpha -\beta }{2} )$

• $\\\tt sin \ \alpha- sin\ \beta = 2cos(\dfrac{\alpha + \beta}{2} )sin (\dfrac{\alpha -\beta }{2} )$