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

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SUMMARY

The difference between cos² and sin² is a well-known trigonometric identity expressed as cos²(a) - sin²(a) = cos(2a). This identity is confirmed by Shaum's outline and is part of a broader set of identities, including the general formula cos(x + y) = cos(x)cos(y) - sin(x)sin(y). By substituting x = y = a, one derives cos(2a) = cos²(a) - sin²(a). Additionally, the identity cos²(a) + sin²(a) = 1 is fundamental in trigonometry, linking to Pythagorean principles.

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  • Understanding of basic trigonometric identities
  • Familiarity with the Binomial Theorem
  • Knowledge of complex numbers and de Moivre's Theorem
  • Basic algebra skills for manipulating trigonometric expressions
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  • Study the derivation of trigonometric identities using the Binomial Theorem
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Students of mathematics, educators teaching trigonometry, and anyone interested in deepening their understanding of trigonometric identities and their applications.

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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!
 
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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).
 
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
 

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