How do you express f(theta)=-cos^2(theta)-7sec^2(theta)-5sec^4theta in terms of non-exponential trigonometric functions?

1 Answer
Mar 15, 2017

see below

Explanation:

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)