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# If sec theta + tan theta = P, then what is cos theta equal to?

Nov 27, 2016

Given $\sec \theta + \tan \theta = P \ldots . \left(1\right)$

we know

${\sec}^{2} \theta - {\tan}^{2} \theta = 1. \ldots \ldots . \left(2\right)$

Dividing (2) by (1) we get

$\sec \theta - \tan \theta = \frac{1}{P} \ldots . \left(3\right)$

Now adding (1) and (3) we have

$2 \sec \theta = P + \frac{1}{P} = \frac{{P}^{2} + 1}{P}$

$\implies \sec \theta = \frac{{P}^{2} + 1}{2 P}$

$\implies \cos \theta = \frac{2 P}{{P}^{2} + 1}$

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