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\(\frac{cot^3\theta}{cosec^2\theta}+\frac{tan^3\theta}{sec^2 \theta}+2sin \theta cos\theta=?\)
1. sin2θcosθ
2. sinθcosθ
3. cosec2θsec2θ
4. cosecθsecθ

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Correct Answer - Option 4 : cosecθsecθ

Given:

⇒ ? = (cot3θ)/(cosec2θ) + (tan3θ)/(sec2θ) + 2sinθcosθ 

Formula:
⇒ sin θ = 1/cosecθ 

⇒ cos θ = 1/secθ 

⇒ cot θ = cos θ /sin θ 

⇒ tan θ = sin θ /cos θ 

⇒ (a + b)2 = a2 + b2 + 2ab

⇒ sin2θ + cos2θ = 1

Calculation:

⇒ ? = (cot3θ)/(cosec2θ) + (tan3θ)/(sec2θ) + 2sinθcosθ 

⇒ ? = cos3θ/sinθ + sin3θ/cosθ + 2sinθcosθ 

⇒ ? = (cos4θ + sin4θ)/sinθ.cosθ + 2sinθ.cosθ 

⇒ ? = (cos4θ + sin4θ + 2sin2θ.cos2θ)/sinθ.cosθ

⇒ ? = (sin2θ + cos2θ)2/(sinθ.cosθ)

⇒ ? = 1/sinθ.cosθ

⇒ ? = cosecθ.secθ

∴ (cot3θ)/(cosec2θ) + (tan3θ)/(sec2θ) + 2sinθcosθ  = cosecθ.secθ

 

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