How do you simplify #cos8theta-5tan2theta# to trigonometric functions of a unit #theta#?

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Answer 1 · 2016-04-16T04:12:01

#cos 8theta-5 tan 2theta=#
#{2[2(2 cos^2 theta-1)^2-1]^2-1}-(10tan theta)/(1-tan^2 theta)#

Use double angle formulas for tangent

#tan 2A= (2 tan A)/(1-tan^2 A)#

Use half-angle formulas for cosine

#cos (A/2)=sqrt((1+cos A)/2#

so that

#cos A=2*cos^2 (A/2)-1#

We can now write

#cos 8theta=2 cos^2 4theta-1#

#cos 4theta=2 cos^2 2theta-1#

#cos 2theta=2 cos^2 theta-1#

It follows that

#cos 8theta-5 tan 2theta=#
#{2[2(2 cos^2 theta-1)^2-1]^2-1}-(10tan theta)/(1-tan^2 theta)#

God bless...I hope the explanation is useful.