Geometry (2)

Geometry Level 3

If sin ⁡ 4 x 2 + cos ⁡ 4 x 3 = 1 5 \dfrac{\sin^4 x}2 + \dfrac{\cos^4 x}3 = \dfrac15 , find 2016 tan ⁡ 2 x 2016\tan^2 x .


The answer is 1344.

Rishabh Deep Singh by Rishabh Deep Singh , 23, India

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1 solution

Chew-Seong Cheong
Jan 20, 2016

sin ⁡ 4 x 2 + cos ⁡ 4 x 3 = 1 5 15 sin ⁡ 4 x + 10 cos ⁡ 4 x = 6 15 ( 1 − cos ⁡ 2 x ) 2 + 10 cos ⁡ 4 x = 6 15 ( 1 − 2 cos ⁡ 2 x + cos ⁡ 4 x ) + 10 cos ⁡ 4 x = 6 25 cos ⁡ 4 x − 30 cos ⁡ 2 x + 9 = 0 ( 5 cos ⁡ 2 x − 3 ) 2 = 0 ⇒ cos ⁡ 2 x = 3 5 ⇒ sin ⁡ 2 x = 1 − cos ⁡ 2 x = 1 − 3 5 = 2 5 ⇒ 2016 tan ⁡ 2 x = 2016 × sin ⁡ 2 x cos ⁡ 2 x = 2016 × 2 3 = 1344 \begin{aligned} \frac{\sin^4 x}{2} + \frac{\cos^4 x}{3} & = \frac{1}{5} \\ 15\sin^4 x + 10\cos^4 x & = 6 \\ 15(1-\cos^2 x)^2 + 10\cos^4 x & = 6 \\ 15(1-2\cos^2 x + \cos^4 x) + 10\cos^4 x & = 6 \\ 25 \cos^4 x - 30 \cos^2 x + 9 & = 0 \\ (5 \cos^2 x - 3)^2 & = 0 \\ \Rightarrow \cos^2 x & = \frac{3}{5} \\ \Rightarrow \sin^2 x & = 1 - \cos^2 x = 1 - \frac{3}{5} = \frac{2}{5} \\ & \\ \Rightarrow 2016 \tan^2 x & = 2016 \times \frac{\sin^2 x}{\cos^2 x} \\ & = 2016 \times \frac{2}{3} \\ & = \boxed{1344} \end{aligned}

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