Tricky Trigonometry - 2

Geometry Level 4

If cot ⁡ ( θ − α ) \cot(\theta - \alpha) , 3 cot ⁡ θ 3 \cot \theta , and cot ⁡ ( θ + α ) \cot(\theta + \alpha) are in arithmetic progression and θ \theta is not an integral multiple of π 2 \dfrac{\pi}{2} , then find the value of sin ⁡ 2 θ sin ⁡ 2 α \dfrac{\sin^2\theta}{\sin^2\alpha} .

Also try Tricky Trigonometry-1 and Tricky Trigonometry-3


The answer is 1.5.

Ram Mohith by Ram Mohith , 18, India

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1 solution

From the given conditions we get c o t ( θ − α ) + c o t ( θ + α ) = 6 c o t θ cot(\theta-\alpha) +cot(\theta+\alpha) =6cot\theta . Simplifying this we get t a n 2 α = tan^2\alpha= 2 t a n 2 θ 3 + t a n 2 θ \dfrac{2tan^2\theta}{3+tan^2\theta} or s i n 2 θ s i n 2 α = 1.5 \dfrac{sin^2\theta}{sin^2\alpha}=\boxed{1.5}

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