Cos^2 - Sin^2 = Cos(2a): Shaum's Solution

[Ai52487963]

Anyone know if the difference of cos^2 and sin^2 is some obscure identity that no one's heard of?

Edit: nevermind. Shaum's tells me that cos^2 - sin^2 = cos(2a). GO SHAUMS!

 

[HallsofIvy]

So the answer to your question is "NO"! It is, in fact, a well known identity!

A more general identity is cos(x+ y)= cos(x)cos(y)- sin(x)sin(y). Letting x= y= a in that, cos(2a)= cos²(a)- sin²(a).

 

[Invictious]

If you know complex numbers [including de Moivre's theorem (great chap, wasn't he) and Binomial Theorem], you can find the exact angle of any sin/cos/tan in surd form, and you can prove any double angle/triple angle/quadruple angle etc angles.

Back on topic though, yes those identities are very famous, and cos^2+sin^2 = 1 is also a darned famous one, think Pythagoras. From that you can get 2 more forms, one with sec and the other with csc