# How do you express cos^3theta+sec^2theta  in terms of non-exponential trigonometric functions?

Nov 27, 2016

${\cos}^{3} \theta + {\sec}^{2} \theta$

$= \frac{1}{4} \left(\cos 3 \theta + 3 \cos \theta\right) + \frac{2}{2 {\cos}^{2} \theta}$

$= \frac{1}{4} \left(\cos 3 \theta + 3 \cos \theta\right) + \frac{2}{1 + \cos 2 \theta}$

$= \frac{\cos 3 \theta + \cos \theta + \cos 3 \theta \cos 2 \theta + 3 \cos \theta \cos 2 \theta + 8}{4 \left(1 + \cos 2 \theta\right)}$

$= \frac{2 \cos 3 \theta + 2 \cos \theta + 2 \cos 3 \theta \cos 2 \theta + 6 \cos \theta \cos 2 \theta + 16}{8 \left(1 + \cos 2 \theta\right)}$

$= \frac{2 \cos 3 \theta + 2 \cos \theta + \cos 5 \theta + \cos \theta + 3 \cos \theta + 3 \cos 3 \theta + 16}{8 \left(1 + \cos 2 \theta\right)}$

$= \frac{5 \cos 3 \theta + 6 \cos \theta + \cos 5 \theta + 16}{8 \left(1 + \cos 2 \theta\right)}$
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