# Evaluate the integral: {eq}\displaystyle \int \frac {1}{cos \ \theta - 1} \ d \theta {/eq}.

## Question:

Evaluate the integral:

{eq}\displaystyle \int \frac {1}{cos \ \theta - 1} \ d \theta {/eq}.

## Double Cosine

The trigonometric identity for the double cosine of an angle {eq}x {/eq} can be expressed in terms of the sine and the cosine of the full angle as,

{eq}\cos 2x=\cos^2 x-\sin^2 x {/eq}.

Moreover, mixing this result with the fundamental trigonometric identity {eq}\sin^2 x+\cos^2 x=1 {/eq} allows us to eliminate one of the trigonometric functions (either the sine or the cosine),

{eq}\cos 2x=1-2\sin^2 x=2\cos^2 x-1 {/eq}.

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To compute the integral,

{eq}I= \displaystyle \int \dfrac{d\theta}{\cos \theta -1} {/eq},

we employ the trigonometric identity,

{eq}\cos...