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# Prove cos(a+b)cos(a-b)=cos^2b-sin^2a ?

Then teach the underlying concepts
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?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

69
Apr 22, 2017

See proof below

#### Explanation:

We need

$\left(x + y\right) \left(x - y\right) = {x}^{2} - {y}^{2}$

$\cos \left(a + b\right) = \cos a \cos b - \sin a \sin b$

$\cos \left(a - b\right) = \cos a \cos b + \sin a \sin b$

${\cos}^{2} a + {\sin}^{2} a = 1$

${\cos}^{2} b + {\sin}^{2} b = 1$

Therefore,

$L H S = \cos \left(a + b\right) \cos \left(a - b\right)$

$= \left(\cos a \cos b - \sin a \sin b\right) \left(\cos a \cos b + \sin a \sin b\right)$

$= {\cos}^{2} a {\cos}^{2} b - {\sin}^{2} a {\sin}^{2} b$

$= {\cos}^{2} b \left(1 - {\sin}^{2} a\right) - {\sin}^{2} a \left(1 - {\cos}^{2} b\right)$

$= {\cos}^{2} b - \cancel{{\cos}^{2} b {\sin}^{2} a} - {\sin}^{2} a + \cancel{{\cos}^{2} b {\sin}^{2} a}$

$= {\cos}^{2} b - {\sin}^{2} a$

$= R H S$

$Q E D$

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