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Identities

Prove the given identity - Please explain in detail.

14. 2 cos x - 2 cos^3x = sin x sin 2x

16. cos (x-y)/cos x cosy = 1 tan x tany

20. sec x -cos x = sin x tanx

22. sin 2x = 1/tanx +cot 2x

23. If tan x = 5/12 and sin x>0 , find sin 2x

26. If sin x = -12/13 with pie <x<3pie/2, and sec y = 13/12 with 3 pie/2 < y< 2 pie, find cos (x+y)

28. If sin x = -2/5 and 3pie/2 < x< 2 pie, then cos (pie/4+x) = ?

Solution Preview

14. 2 cos x - 2 cos^3x = sin x sin 2x
Proof: 2 cos x - 2 cos^3x =2cosx(1-cos^2x)=2cosxsin^2x=(2sinxcosx)sinx=sin2xsinx
Thus 2 cos x - 2 cos^3x =sinx sin2x

16. cos (x-y)/cos x cosy = 1+ tan x tany
Proof: cos (x-y)/cos x cosy=(cosxcosy+sinxsiny)/cosxcosy=1+sinxsiny/cosxcosy
=1+tanx tany
Thus cos (x-y)/cos x cosy=1+tanx ...

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