Trigonometry/Multiple-Angle and Product-to-sum Formulas

Multiple-Angle Formulas

• $\sin (2a)=2\sin a\cos a$
• $\cos (2a)={\cos }^{2}a-{\sin }^{2}a=1-2{\sin }^{2}a=2{\cos }^{2}a-1$
• $\tan (2a)=\frac{2\tan a}{1-{\tan }^{2}a}$

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Proofs for Double Angle Formulas

Using the Sum and Difference Identities

Recall that:
$\cos (a+b)=\cos (a)\cos (b)-\sin (a)\sin (b)$

Using a = b in the above formula yields:

  $\cos (2a)=\cos (a+a)=\cos (a)\cos (a)-\sin (a)\sin (a)={\cos }^2(a)-{\sin }^2(a)$

From the last (rightmost) term, two more identities may be derived. One containing only a sine:

  $\cos (2a)={\cos }^2(a)-{\sin }^2(a)={\cos }^2(a)-{\sin }^2(a)+0={\cos }^2(a)-{\sin }^2(a)+[{\sin }^2(a)-{\sin }^2(a)]=[{\cos }^2(a)+{\sin }^2(a)]-2{\sin }^2(a)=1-2{\sin }^2(a)$

And one containing only a cosine:

  $\cos (2a)={\cos }^2(a)-{\sin }^2(a)={\cos }^2(a)-{\sin }^2(a)+0={\cos }^2(a)-{\sin }^2(a)+[{\cos }^2(a)-{\cos }^2(a)]=2{\cos }^2(a)-[{\cos }^2(a)+{\sin }^2(a)]=2{\cos }^2(a)-1$

$\sin (a+b)=\sin (a)\cos (b)+\sin (b)\cos (a)$ ; a = b for sin(2a)

  $\sin (a)\cos (a)+\sin (a)\cos (a)=2\cos (a)\sin (a)=\sin (2a)$