# How do you verify the identity (tanx+secx)(1-sinx)=cosx?

Aug 26, 2016

See below.

#### Explanation:

Use the following identities to simplify the left-hand side:

$\tan \beta = \sin \frac{\beta}{\cos} \beta$

$\sec \beta = \frac{1}{\cos} \beta$

Start the simplification process:

$\left(\sin \frac{x}{\cos} x + \frac{1}{\cos} x\right) \left(1 - \sin x\right) = \cos x$

$\left(\frac{\sin x + 1}{\cos x}\right) \left(1 - \sin x\right) = \cos x$

$\frac{1 - {\sin}^{2} x}{\cos} x = \cos x$

Now, rearrange the pythagorean identity ${\sin}^{2} \beta + {\cos}^{2} \beta = 1$ for ${\cos}^{2} \beta$ to get ${\cos}^{2} \beta = 1 - {\sin}^{2} \beta$. Applying this to our problem:

${\cos}^{2} \frac{x}{\cos} x = \cos x$

$\frac{\left(\cos x\right) \left(\cos x\right)}{\cos} x = \cos x$

$\frac{\cancel{\cos x} \cos x}{\cancel{\cos x}} = \cos x$

$\cos x = \cos x$

Identity proved!!

Hopefully this helps!