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

Question

2704 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-26T19:43:25

see below

Left Hand Side:
$=(sin^4beta-2sin^2beta+1)cosbeta$

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

The factors are:
$(sin^2 beta-1)(sin^2beta-1) = (sin^2beta-1)^2$

Hence,
$=(sin^2 beta -1)^2 cos beta$

$=[-1(1-sin^2 beta)]^2 cos beta$

$=(1-sin^2beta)^2cos beta$ ; Use identity $sin^2 beta + cos^2 beta =1$

$=(cos^2 beta)^2 cos beta$

$=cos^4 beta cos beta$

$=cos^5 beta$

$:. =$ Right Hand Side

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