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By expressing cos(2x) in terms of cos(x) find the exact value of the integral of cos(2x)/cos^2(x) between the bounds pi/4 and pi/3.

cos(2x)=cos²(x)-sin²(x)=2cos²(x)-1
Therefore:cos(2x)/cos²(x)=(2cos²(x)-1)/cos²(x)=2cos²(x)/cos²(x) - 1/cos²(x)=2 - 1/cos²(x)=2 - sec²(x)
Integral of sec²(x) = tan(x)
Integral of 2 = 2x
[2x - tan(x)] between pi/4 and pi/3
= (2pi/3 - tan(pi/3)) -(pi/2 - tan(pi/4))
= (2pi/3 - sqrt(3)) - (pi/2 - 1)
= pi/6 - sqrt(3) + 1

Answered by Hugo F. • Maths tutor

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By expressing cos(2x) in terms of cos(x) find the exact value of the integral of cos(2x)/cos^2(x) between the bounds pi/4 and pi/3.

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