If 2x = sec A and \frac{2}{x} = tanA then 2 \left(\right. x^{2}

Foundation

Mathematics Foundation

Introduction to Trigonometry

Question

If 2x = sec A and $\frac{2}{x} = tanA$ then $2 (x^{2} - \frac{1}{x^{2}}) = ?$

$\frac{1}{16}$

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{8}$

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Given: 2x = sec A and 2/x = tan A. We need to find 2(x² - 1/x²).

Recall the identity: sec²A - tan²A = 1.

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

This simplifies to: ${(2 x)}^{2} - {(\frac{2}{x})}^{2} = 1$

Which becomes: $4 x^{2} - \frac{4}{x^{2}} = 1$

Divide both sides by 2: $2 x^{2} - \frac{2}{x^{2}} = \frac{1}{2}$

This is equivalent to: $2 (x^{2} - \frac{1}{x^{2}}) = \frac{1}{2}$

Final Answer: $\frac{1}{2}$