Question

$\huge{\underline{\underline{\bf Question}}}$

$if \sf 2x = SecA \: and \: \dfrac{2}{x} = tanA$ find the value of :-

$\sf 2 \bigg(x²-\dfrac{1}{x²}\bigg)$

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Step by step explanation ✅

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Answer 1

Answer:

As we know,

➺ sec²θ−tan²θ=1

Substitute the given values in terms of x.

➺ (2x)²−(2/x)² = 1

➺ 4x²− 4/x² = 1

➺ 4(x²–1/x²) = 1

➺ x²–1/x² = 1/4

Answer 2

Given :

• $\sf 2x = sec A$
• $\sf \dfrac{2}{x} = tan A$

To Find :

• $\sf 2 \bigg(x²-\dfrac{1}{x²}\bigg)$

Solution :

$\Rightarrow \sf sec A = 2x$$\Rightarrow \sf sec^2 A = 4x^2$

$\Rightarrow\sf tan A = \dfrac{2}{x}$$\Rightarrow \sf tan^2 A = \dfrac{4}{x^2}$

Identity to be used,

$\sf tan^2 A +1 = sec ^2 A$

Substituting the values,

$\longrightarrow \sf \dfrac{4}{x^2}+1=4x^2\\\\\\\longrightarrow \sf 1 = 4x^2 - \dfrac{4}{x^2}\\\\\\\longrightarrow \sf 1 = 4 \Big(x^2 - \dfrac{1}{x^2}\Big)\\\\\\\longrightarrow \sf \dfrac{1}{2} = 2 \Big ( x^2 - \dfrac{1}{x^2}\Big)$

Required Answer :

$\boxed{\boxed{\sf 2 \bigg(x²-\dfrac{1}{x²}\bigg)= \dfrac{1}{2}}}$