Verifying Trigonometric Identities

[Eliotmason]

Been stuck on this question for a while now and can't seem to get anywhere:

Verify the identity:

\(\displaystyle \sin ^{4}\theta -\cos ^{4}\theta =2\sin ^{2}\theta-1\)

I started from the left and I only got to here

\(\displaystyle (\sin\theta-\cos\theta)(\sin\theta+\cos\theta)(\sin^{2}\theta+\cos^{2}\theta)\)

I really don't even know if I am on the right track. Any help would be great! Thank you. :razz:

 

[stapel]

Verify the identity:

\(\displaystyle \sin ^{4}\theta -\cos ^{4}\theta =2\sin ^{2}\theta-1\)

I started from the left and I only got to here

\(\displaystyle (\sin\theta-\cos\theta)(\sin\theta+\cos\theta)(\sin^{2}\theta+\cos^{2}\theta)\)

The third factor on the left-hand side is just 1, so now the equation is:

(sin(@) - cos(@))(sin(@) + cos(@)) = 2sin^2(@) - 1

Multiply the two factors on the left-hand side back together, and note that sin^2(@) - cos^2(@) = sin^2(@) - [1 - cos^2(@)].

 

[Eliotmason]

The third factor on the left-hand side is just 1, so now the equation is:

(sin(@) - cos(@))(sin(@) + cos(@)) = 2sin^2(@) - 1

Multiply the two factors on the left-hand side back together, and note that sin^2(@) - cos^2(@) = sin^2(@) - [1 - cos^2(@)].

:shock: Great Scott! Its Magic!

Stapel, you are an ABSOLUTE GENIUS!!! Thank you so much for your help.