sin 2tsin4tcos2tcos4tSolutionSolution first the numerator si

(sin 2t+sin4t)/(cos2t-cos4t)

Solution

Solution:

first the numerator
(sin4t+sin2t)
=sin(2t+2t) + sin2t
=sin2t cos2t + cos2t sin2t + 2sint cost
=2 sin2t cos2t + 2sint cost
=2(2cost sint)(cos^2 t-sin^2 t) + 2sint cost
=(4cos^3 t)(sin t) - (4cos t)(sin^3 t) + 2sint cost
=2sint cos t (2cos^2 t - 2sin^2 t +1)
=2sint cos t(2cos^2 t - sin^2 t - sin^2 t +1)
=2sint cos t(2cos^2 t - sin^2 t + cos^2 t)
=2sint cos t(3cos^2 t - sin^2 t )

now lets do the denominator

cos2t - cos4t
= cos2t - {cos2tcos2t - sin2tsin2t}
= cos2t - (cos2t)^2 + (sin2t)^2
= cos2t (1 - cos2t) + (sin2t)^2
= cos2t (1 - (cos^2 t - sin^2 t)) + 4cos^2 t sin^2 t
= (cos^2 t -sin^2 t)(2sin^2 t) + 4cos^2 t sin^2 t
=(2sin^2 t cos^2 t) - 2 sin^4 t + 4cos^2 t sin^2 t
= -2 sin^4 t + 6cos^2 t sin^2 t
=(2sin^2 t)(3 cos^2 t - sin^2 t)

now let put the denominator and numerator together

(sin 2t+sin4t)/(cos2t-cos4t) = {2sint cos t(3cos^2 t - sin^2 t)} / {(2sin^2 t)(3 cos^2 t - sin^2 t)}

                                        = cost / sint = cot t