Question

prove that cos 20 cos 40 cos 80= 1/8

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Answer 1

Answer:-

$\mathsf{cos20.cos40.cos80 = \dfrac{1}{8 } }$

Explanation:-

To Prove

cos 20° cos 40° cos80° = $\dfrac{1}{8}$

Proof:

Consider LHS

$\mathsf{ cos20. cos40. cos80}$

On dividing and multipling with 2 Sin 20°

$\mathsf{= \dfrac{1}{2 sin20} (2 sin20. cos20. cos40. cos80)}$

$\mathsf{= \dfrac{1}{2 sin20} ( sin40.cos40. cos80)}$

On dividing and multiplying with 2

$\mathsf{= \dfrac{1}{4 sin20} (2 sin40.cos40. cos80)}$

$\mathsf{= \dfrac{1}{4 sin20} ( sin80. cos80)}$

On dividing and multiplying with 2

$\mathsf{= \dfrac{1}{8 sin20} ( 2 sin80. cos80)}$

$\mathsf{= \dfrac{1}{8 sin20} ( sin 160)}$

$\mathsf{= \dfrac{1}{8 sin20}[sin (180-20)]}$

$\mathsf{= \dfrac{1}{8 sin20} ( sin 20)}$

$\mathsf{= \dfrac{1}{8\cancel{ sin20}} ( \cancel{sin 20})}$

$\bold{= \dfrac{1}{8 } }$

RHS

$\bold{= \dfrac{1}{8 } }$

Hence proved!

Identities used:

• 2 sinx cosx = sin2x

• sin(180-x) = sinx