Prove that: cos 78 ° cos 42 ° cos 36 ° = 1 8 - Mathematics

Question

Prove that: $\cos 78 ° \cos 42 ° \cos 36 ° = \frac{1}{8}$

Numerical

Solution

$L H S = \cos 78 ° \cos 42 ° \cos 36 °$
$= \frac{(2 \cos 78 ° \cos 42 °)}{2} \cos 36 °$
$= \frac{\cos (78 ° + 42 °) + \cos (78 ° - 42 °)}{2} \times \cos 36 °$
$[2 cos A cos B = \cos (A + B) + \cos (A - B)]$
$= \frac{1}{2} (\cos 120 ° + \cos 36 °) \cos 36 °$

$= \frac{1}{2} (- \cos (180 ° - 120 °) + \cos 36 °) \cos 36 °$
$= \frac{1}{2} (- \cos 60 ° + \cos 36 °) \cos 36 °$
$= \frac{1}{2} (- \frac{1}{2} + \frac{\sqrt{5} + 1}{4}) \frac{\sqrt{5} + 1}{4}$
$= \frac{1}{2} \times \frac{\sqrt{5} - 1}{4} \times \frac{\sqrt{5} + 1}{4}$
$= \frac{1}{8}$
$= R H S$
$Hence proved .$

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Values of Trigonometric Functions at Multiples and Submultiples of an Angle

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Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.3 [Page 42]

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Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.3 | Q 4 | Page 42

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Prove that: cos 78 ° cos 42 ° cos 36 ° = 1 8 - Mathematics

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