The value of ( 2cos⁡³θ−cos⁡θover sin⁡θ−2sin⁡³θ )² +1, 0 ≠ 45° is:

The value of ( 2cos⁡³θ−cos⁡θover sin⁡θ−2sin⁡³θ )² +1, 0 ≠…

The value of $\left( 2cos⁡^3θ−cos⁡θ\over sin⁡θ−2sin⁡^3θ \right)^2$ +1, 0 ≠ 45° is:

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Correct answer: C

Explanation

The correct answer is option 3 i.e. cosec²θ.

⇒ $\left( 2cos⁡^3θ−cos⁡θ\over sin⁡θ−2sin⁡^3θ \right)^2$ + 1

⇒ $\left( (2cos⁡^2θ−1)cos⁡θ\over sin⁡θ(1−2sin⁡^2θ) \right)^2$ + 1

⇒ $\left( (cos⁡2θ)cos⁡θ\over sin⁡θ(Cos2θ) \right)^2$ + 1

⇒ $\left( cos⁡θ\over sin⁡θ \right)^2$ + 1

⇒ $⁡sec^2θ$ + 1

⇒ cosec²θ