Use the Double Argument Property
cos 2 theta=2cos^2 theta -1---> If we solve for cos^2 theta then we have color(red)(cos^2 theta =1/2 (cos 2theta+1))
cos 4 theta=2cos^2 2theta -1
=2(cos 2 theta)^2-1
=2(2cos^2theta-1)^2-1
=2(4cos^4 theta-4 cos^2 theta +1)-1
=8cos^4 theta-8cos^2 theta +2-1
=8cos^4 theta-8cos^2 theta +1--> If we put in 1/2 (cos 2theta+1) for cos^2 theta we have
=8cos^4 theta-8(1/2 (cos 2 theta+1)) +1
=8cos^4 theta-4 (cos 2 theta+1) +1
=8cos^4 theta-4 cos 2 theta-4 +1
=8cos^4 theta-4 cos 2 theta-3---> Now solve for cos^4 theta
color(red)(1/ 8 (cos 4 theta+4cos 2 theta+3)= cos^4 theta
Therefore,
f(theta)=-cos^2 theta-7 sec^2 theta-5 sec^4 theta
=-cos^2 theta -7/cos^2 theta - 5 / cos^4 theta
=-1/2 (cos 2theta+1)-7/(1/2 (cos 2theta+1)) - 5 /(1/ 8 (cos 4 theta+4cos 2 theta+3))
:.color(blue)(f(theta)=-1/2 (cos 2theta+1)-14/(cos 2theta+1)- 40/(cos 4 theta+4cos 2 theta+3)