Prove: sin 5x + sin 4x + sin 2x +sinx/cos 5x +cos 4x + cos2x + cosx = tan3x
Answers
Refer to the attachment because it is similar!
Given : (sin 5x + sin 4x + sin 2x +sinx)/(cos 5x +cos 4x + cos2x + cosx) = tan3x
To Find : Prove
Solution:
sin C + sinD = 2sin{(C + D)/2}Cos{(C- D)/2}
cos C + cosD = 2cos{(C + D)/2}Cos{(C- D)/2}
LHS = (sin 5x + sin 4x + sin 2x +sinx)/(cos 5x +cos 4x + cos2x + cosx)
Numerator
= sin 5x + sin 4x + sin 2x +sinx
= (sin 5x + sinx) + (sin4x +sin2x)
= 2sin3xcos2x + 2sin3xcosx
= 2sin3x( cos2x + cosx)
Denominator
= cos 5x +cos 4x + cos2x + cosx
= (cos5x + cosx) + ( cos4x + cos2x)
= 2cos3xcos2x + 2cos3xcosx
= 2cos3x( cos2x + cosx)
LHS = (2sin3x( cos2x + cosx))/ (2cos3x( cos2x + cosx))
= sin3x/cos3x
= tan 3x
= RHS
QED
Hence proved
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