# How do you verify the identity (sin^4beta-2sin^2beta+1)cosbeta=cos^5beta?

Jan 26, 2017

see below

#### Explanation:

Left Hand Side:
$= \left({\sin}^{4} \beta - 2 {\sin}^{2} \beta + 1\right) \cos \beta$

Note that ${\sin}^{4} \beta - 2 {\sin}^{2} \beta + 1$ looks like ${x}^{4} - 2 {x}^{2} + 1$ so we can factor it.

The factors are:
$\left({\sin}^{2} \beta - 1\right) \left({\sin}^{2} \beta - 1\right) = {\left({\sin}^{2} \beta - 1\right)}^{2}$

Hence,
$= {\left({\sin}^{2} \beta - 1\right)}^{2} \cos \beta$

$= {\left[- 1 \left(1 - {\sin}^{2} \beta\right)\right]}^{2} \cos \beta$

$= {\left(1 - {\sin}^{2} \beta\right)}^{2} \cos \beta$; Use identity ${\sin}^{2} \beta + {\cos}^{2} \beta = 1$

$= {\left({\cos}^{2} \beta\right)}^{2} \cos \beta$

$= {\cos}^{4} \beta \cos \beta$

$= {\cos}^{5} \beta$

$\therefore =$ Right Hand Side