Question

50 points pls answer it fast​

Answer 1

Answer :-

The value of 2(x² - 1/x²) is 1/2.

Explanation :-

Given :-

• 2x = sec θ

• 2/x = tan θ

To find :-

2(x² - 1/x²)

Solution :-

2x = sec θ ---eq(1)

2/x = tan θ ---eq(2)

Adding (1) and (2)

2x + 2/x = sec θ + tan θ

2(x + 1/x) = sec θ + tan θ

x + 1/x = (sec θ + tan θ)/2 ---eq(3)

Subtracting (2) from (1)

2x - 2/x = sec θ - tan θ

2(x - 1/x) = sec θ - tan θ

x - 1/x = (sec θ - tan θ)/2 --eq(4)

Multiplying (3) and (4)

$\sf \bigg(x + \dfrac{1}{x} \bigg) \bigg(x - \dfrac{1}{x} \bigg) = \bigg( \dfrac{\sec\theta + \tan\theta}{2}\bigg)\bigg(\dfrac{\sec\theta - \tan\theta}{2}\bigg) \\ \\ \sf {x}^{2} - \dfrac{1}{ {x}^{2} } = \dfrac{\sec^2\theta - \tan^2\theta}{4} \\ \\ \bf \because (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ \sf \sf {x}^{2} - \dfrac{1}{ {x}^{2} } = \dfrac{1}{4} \\ \\ \boxed{\bf \because \sec^2\theta - \tan^2\theta = 1}$

Multiplying by 2 on both sides

$\\ \\ \sf 2 \bigg({x}^{2} - \dfrac{1}{ {x}^{2} } \bigg)= 2\bigg(\dfrac{1}{4}\bigg) \\ \\ \sf 2 \bigg({x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = \dfrac{1}{2}$

Therefore the value of 2(x² - 1/x²) is 1/2.

Answer 2

Answer: 1/2

Step-by-step explanation:

Given,

2x = secФ

x = secФ/2

Making square of both sides,

x² = (sec²Ф)/4        ..................................I)

And also,

2/x = tanФ

1/x = (tanФ)/2

Making square of both sides,

1/x² = (tan²Ф)/4    .................................II)

Now, Subtracting equation II) from I)

We have,

x² - 1/x² = (sec²Ф)/4 - (tan²Ф)/4

x² - 1/x² = (sec²Ф - tan²Ф)/4

x² - 1/x² = 1/4             (∵sec²Ф - tan²Ф = 1)

Multiplying with 2 on both sides

2(x² - 1/x²) = 2/4

2(x² - 1/x²) = 1/2