If θ is a positive acute angle and 3 (\sec^2\theta + \tan^2\theta) = 5, then the value of cos 2θ is

SSC CGL Tier II 12 Apr 2015 Paper

Question : 43 of 100

Marks: +1, -0

If θ is a positive acute angle and

3 (sec^2θ + tan^2θ) = 5

, then the value of cos 2θ is

Solution:

(1)

3 (\sec ^{2} \θ+\tan ^{2}\θ)=5

⇒ \sec ^{2} \θ+\tan ^{2} \θ= {5}/{3}

⇒ \sec ^{2}\θ+\sec ^{2}\θ-1= {5}/{3}

⇒ 2 \sec ^{2}\θ= {5}/{3}+1= {8}/{3}

⇒ \sec^{2}\θ= {4}/{3} ⇒ \sec \θ= {2}/{√{3}}

⇒ \cos \θ= {√{3}}/{2}=\cos 30°

⇒\θ=30°

∴ \cos 2 \θ=\cos 60°= {1}/{2}

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