How do you express #f(theta)=-2sin^2(theta)-9sec^2(theta)-tan^4theta# in terms of non-exponential trigonometric functions?

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Answer 1 · 2018-02-17T13:56:29

Let me first represent #f(theta)# in form of sines and cosines.

#f(theta)=-2sin^2(theta)-9sec^2(theta)-tan^4(theta)#

#=>f(theta)=-2sin^2(theta)-9/cos^2(theta)-(sin^2(theta)/cos^2(theta))^2#

Next use identities to find that,

#sin^2(theta)=(1-cos(2theta))/2#

and

#cos^2(theta)=(1+cos(2theta))/2#

Use these to solve for #f(theta)#,

#f(theta)=cos(2theta)-1-18/(1+cos(2theta))-((1-cos(2theta))/(1+cos(2theta)))^2#

Simplify it if needed.

Somewhere, #cos^2(2theta)# will appear. For that, substitute

#cos^2(2theta)=(1+cos(4theta))/2#