(Sin theta+ sin3theta + sin5theta)÷ (cos theta + cos3theta +cos 5theta
To simplify the given expression, we can use the trigonometric identity:
sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2)
Therefore, we have:
sin(theta) + sin(3theta) = 2sin((theta+3theta)/2)cos((3theta-theta)/2)
= 2sin(2theta)cos(theta)
Likewise:
sin(3theta) + sin(5theta) = 2sin((3theta+5theta)/2)cos((5theta-3theta)/2)
= 2sin(4theta)cos(theta)
And:
sin(5theta) + sin(theta) = 2sin((5theta+theta)/2)cos((5theta-theta)/2)
= 2sin(3theta)cos(2theta)
Plugging these results back into the original expression:
(sin(theta) + sin(3theta) + sin(5theta)) / (cos(theta) + cos(3theta) +cos(5theta))
= (2sin(2theta)cos(theta) + 2sin(4theta)cos(theta) + 2sin(3theta)cos(2theta)) / (cos(theta) + cos(3theta) +cos(5theta))
= 2(cos(theta)(sin(2theta) + sin(4theta)) + cos(2theta)sin(3theta)) / (cos(theta) + cos(3theta) +cos(5theta))
= 2cos(theta)(2sin(3theta)cos(theta) + cos(2theta)sin(3theta)) / (cos(theta) + cos(3theta) +cos(5theta))
= 2cos(theta)sin(3theta)(2cos(theta) + cos(2theta)) / (cos(theta) + cos(3theta) +cos(5theta))
And this would be the simplified expression for the given trigonometric expression.