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If 4(cosec2 57° - tan2 33°) - cos 90° - y tan2 66° tan2 24° \(=\dfrac{y}{2}\), the value of y is:  
1. \(\dfrac{1}{3}\)
2. \(\dfrac{8}{3}\)
3. \(8\)
4. \(\dfrac{3}{8}\)

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Correct Answer - Option 2 : \(\dfrac{8}{3}\)

Given:

4(cosec2 57° tan2 33°) - cos 90° y tan2 66° tan2 24° \(=\dfrac{y}{2}\)

Formula used:

cosec2θ  cot2θ = 1

tan(90 – θ) = cotθ
Calculation:

4(cosec2 57°  tan2 33°) – cos 90°  y tan2 66° tan2 24° \(=\dfrac{y}{2}\)

⇒ 4[cosec2 57°  cot2(90° – 33°)] – cos 90°  y tan2 66° cot2(90° – 24°) \(=\dfrac{y}{2}\)

⇒ 4(cosec2 57°  cot2 57°) – cos 90°  y tan2 66° cot2 66° \(=\dfrac{y}{2}\)

⇒ 4 – 0 – y = y/2

⇒ 8 = 3y

⇒ y = 8/3

∴ The value of y is 8/3.

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