How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{12})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 12 )$ without using products of trigonometric functions?

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{12})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 12 )$ without using products of trigonometric functions?

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Answer 1 · 2016-02-09T03:42:04

$\frac{(\sqrt{2 - \sqrt{2}}) . (\sqrt{2 + \sqrt{3}})}{4}$ $((sqrt(2 - sqrt2)).(sqrt(2 + sqrt3))]/4$

Explanation:

Product $P = sin (\frac{π}{8}) . cos (\frac{5 π}{12})$ $P = sin (pi/8).cos ((5pi)/12)$ .
a. Find $sin (\frac{π}{8})$ $sin (pi/8)$ by trig identity: $cos 2 x = 1 - 2 {sin}^{2} x$ $cos 2x = 1 - 2sin^2 x$
$cos (\frac{π}{4}) = \frac{\sqrt{2}}{2} = 1 - 2 {sin}^{2} (\frac{π}{8})$ $cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)$
${sin}^{2} (\frac{π}{8}) = \frac{2 - \sqrt{2}}{4}$ $sin^2 (pi/8) = (2 - sqrt2)/4$
$sin (\frac{π}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$ $sin (pi/8) = sqrt(2 - sqrt2)/2$ --> (sin $\frac{π}{8}$ $pi/8$ is positive)
b. Find $cos (\frac{5 π}{12}) = cos t$ $cos ((5pi)/12) = cos t$ by identity: $cos 2 t = 2 {cos}^{2} t - 1$ $cos 2t = 2cos^2 t - 1$
$cos 2 t = cos (\frac{10 π}{12}) = cos (\frac{5 π}{6}) = - \frac{\sqrt{3}}{2}$ $cos 2t = cos ((10pi)/12) = cos ((5pi)/6) = -sqrt3/2$
$\frac{\sqrt{3}}{2} = 2 {cos}^{2} t - 1$ $sqrt3/2 = 2cos^2 t - 1$
$2 {cos}^{2} t = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$ $2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2$
${cos}^{2} t = \frac{2 + \sqrt{3}}{4}$ $cos^2 t = (2 + sqrt3)/4$
$cos t = cos (\frac{5 π}{12}) = \frac{\sqrt{2 + \sqrt{3}}}{2}$ $cos t = cos ((5pi)/12) = sqrt(2 + sqrt3)/2$ --> (cos t is positive)

Finally: $P = \frac{(\sqrt{2 - \sqrt{2}}) . (\sqrt{2 + \sqrt{3}})}{4}$ $P = [(sqrt(2 - sqrt2}).(sqrt(2 + sqrt3))]/4$

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{12})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 12 )$ without using products of trigonometric functions?

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Explanation:

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How do you express $sin (\frac{π}{8}) \cdot cos ((\frac{5 π}{12})$ $sin(pi/ 8 ) * cos(( ( 5 pi) / 12 )$ without using products of trigonometric functions?