Progression in Trigonometry

Geometry Level 2

If sin ⁡ A \sin A , cos ⁡ A \cos A , and tan ⁡ A \tan A are in a geometric progression , then find the value of: cos ⁡ 3 A + cos ⁡ 2 A \large \cos^3 A + \cos^2 A


The answer is 1.

Ram Mohith by Ram Mohith , 18, India

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1 solution

Chew-Seong Cheong
Sep 28, 2018

If sin ⁡ A \sin A , cos ⁡ A \cos A , and tan ⁡ A \tan A are in a geometric progression, then

sin ⁡ A tan ⁡ A = cos ⁡ 2 A sin ⁡ 2 A cos ⁡ A = cos ⁡ 2 A sin ⁡ 2 A = cos ⁡ 3 A \begin{aligned} \sin A \tan A & = \cos^2 A \\ \frac {\sin^2 A}{\cos A} & = \cos^2 A \\ \sin^2 A & = \cos^3 A \end{aligned}

Then we have cos ⁡ 3 A + cos ⁡ 2 A = sin ⁡ 2 A + cos ⁡ 2 A = 1 \cos^3 A + \cos^2 A = \sin^2 A + \cos^2 A = \boxed 1 .

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