How do you verify sin x(1- 2 cos ^2 x + cos ^4 x) = sin ^5 x?

2 Answers
Aug 3, 2015

Verify sin x(1 - 2cos^2 x + cos^4 x) = sin^5 x

Explanation:

sin^4 x = (sin^2 x)^2 = (1 - cos ^2 x)^2 = (1 - 2cos^2 x + cos^4x)

sin x.sin^4 x = sin^5x = sin x(1 - 2cos^2 x + cos^4 x)

Aug 3, 2015

The key to the proof is to factor the left hand side

Explanation:

When I see a set of parenthesis I always want to see if it can be factored into a simpler form.

So it turns out that (1-2cos^2(x)+cos^4(x)) is a typical form and is factored by the the following form:

(a^2-2ab+b^2) = (a-b)^2

so our factored form is:

(1-2cos^2(x)+cos^4(x))=(1-cos^2(x))^2

Now I see that we have cos^2(x) and that makes me want to check the pythagorean identity:

sin^2(x)+cos^2(x)=1

notice if we move the cosine to the other side then we have the
same expression.

sin^2(x) = 1-cos^2(x)

so let us substitute the sin^2(x) into our equation:
sin(x)(sin^2(x))^2 = sin^5(x)

now it is just a matter of simplifying the left hand side.

sin(x)*sin^4(x) = sin^5(x)
sin^5(x) = sin^5(x)

and thus it is demonstrated.
and that makes us happy.