How do you use the sum or difference identities to find the exact value of $cos (- \frac{π}{12})$ $cos(-pi/12)$ ?

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Answer 1 · 2017-02-09T09:10:48

$cos (- \frac{π}{12}) = \frac{\sqrt{3} + 1}{2 \sqrt{2}}$ $cos(-pi/12)=(sqrt3+1)/(2sqrt2)$

As $\frac{π}{4} - \frac{π}{3} = - \frac{π}{12}$ $pi/4-pi/3=-pi/12$ , we can use here the difference identity for cosine ratio. According to this

$cos (A - B) = cos A cos B + sin A sin B$ $cos(A-B)=cosAcosB+sinAsinB$

Hence $cos (- \frac{π}{12}) = cos (\frac{π}{4} - \frac{π}{3})$ $cos(-pi/12)=cos(pi/4-pi/3)$

= $cos (\frac{π}{4}) cos (\frac{π}{3}) + sin (\frac{π}{4}) sin (\frac{π}{3})$ $cos(pi/4)cos(pi/3)+sin(pi/4)sin(pi/3)$

= $\frac{1}{\sqrt{2}} \times \frac{1}{2} + \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}$ $1/sqrt2xx1/2+1/sqrt2xxsqrt3/2$

= $\frac{\sqrt{3} + 1}{2 \sqrt{2}}$ $(sqrt3+1)/(2sqrt2)$

How do you use the sum or difference identities to find the exact value of cos(−π12)cos(-pi/12)?