If 3x = cosec θ and \frac{3}{x} = cotθ then 3 \left(\right. x^{2}

Foundation

Mathematics Foundation

Introduction to Trigonometry

Question

If 3x = cosec θ and $\frac{3}{x} = cotθ$ then $3 (x^{2} - \frac{1}{x^{2}}) = ?$

$\frac{1}{81}$

$\frac{1}{27}$

$\frac{1}{3}$

$\frac{1}{9}$

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Given: 3x = csc θ and 3/x = cot θ. We need to find 3(x² - 1/x²).

Recall the identity: csc²θ - cot²θ = 1.

Substitute the given expressions: (3x)² - (3/x)² = 1.

This simplifies to: 9x² - 9/x² = 1.

Divide both sides by 3: 3x² - 3/x² = 1/3.

Which is exactly: 3(x² - 1/x²) = 1/3.

Final answer: $\frac{1}{3}$