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← JEE Main 2015 · 11 Apr · Shift 1
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JEE Main 2015 · 11 Apr · Shift 1, Mathematics question 88

If cos⁡α+cos⁡β=32\cos\alpha+\cos\beta=\dfrac32 and sin⁡α+sin⁡β=12\sin\alpha+\sin\beta=\dfrac12 and θ is the arithmetic mean of α and β, then sin⁡2θ+cos⁡2θ\sin2\theta+\cos2\theta is equal to :

Choose one option
Answer and worked solution

Official answer: C

Dividing the sum formulas: tan⁡α+β2=tan⁡θ=13\tan\frac{\alpha+\beta}2=\tan\theta=\frac13.
sin⁡2θ+cos⁡2θ=2t+1−t21+t2=23+1−19109=75.\sin2\theta+\cos2\theta=\frac{2t+1-t^2}{1+t^2}=\frac{\frac23+1-\frac19}{\frac{10}9}=\frac75.

Topics:Trigonometric ratios and identities
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