Achiever Test 2 QR #39

[Decan]

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Hi...quick question about this identity.

The problem in question is (sin^4theta - cos^4theta)/sin2theta

I understand that the numerator can be simplified via difference of squares to (sin^2theta - cos^2theta)(sin^2theta + cos^2theta)/sin2theta and (sin^2theta + cos^2theta) equals 1 so we get (sin^2theta - cos^2theta)/sin2theta

My question is, how does (sin^2theta - cos^2theta) = -cos2theta? I tired to solve this by simplifying (sin^2theta - cos^2theta) by using difference of squares. Is this just an identity that I never learned?

 

[tresbon]

Hi...quick question about this identity.

The problem in question is (sin^4theta - cos^4theta)/sin2theta

I understand that the numerator can be simplified via difference of squares to (sin^2theta - cos^2theta)(sin^2theta + cos^2theta)/sin2theta and (sin^2theta + cos^2theta) equals 1 so we get (sin^2theta - cos^2theta)/sin2theta

My question is, how does (sin^2theta - cos^2theta) = -cos2theta? I tired to solve this by simplifying (sin^2theta - cos^2theta) by using difference of squares. Is this just an identity that I never learned?

(sin^2theta - cos^2theta) = -cos2theta is a trig identity.

Related trig identities to this alone:

cos2theta = 2cos^2theta - 1 or cos2theta = 1 - 2sin^2theta

 

[Decan]

Ah...I see. I just didnt know this identity. Thanks for the clarification!