Quick simple indentity explanation needed

[amd123]

Homework Statement

When you have something like

and you want to use the pythagorean's theorem identity but let's say i do it with the 3rd term in the parenthesis, then would this be valid

and what about this

is this also valid?

Homework Equations

N/A

The Attempt at a Solution

N/A

 

[praharmitra]

no it would not be valid.

to verify this, cube and square sin(x)^2 + cos(x)^2 = 1, and check

 

[yeongil]

Heh, my students just learned proving identities, and they've been kicking them in the ***...
sin^{5}x cos^{2}x = (cos^{2}x - 2cos^{4}x + cos^{6}x)sin x
You do want to use the Pythagorean identity, but not in the first step. You can also start on either side, but I did from the left.

If you start from the left side, my hint would be to rewrite sin^{5}x cos^{2}x using the properties of exponents so that it equals (everything else) times sin x.


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