Please answer this with full explanation. If 5x = sec θ and (5/x) = tan θ, then find the value of: 5( {x}^(2) - \frac{1}{ {x}^(2) } ) ​

Question

Please answer this with full explanation.

If 5x = sec θ and (5/x) = tan θ, then find the value of:


`5( {x}^(2) - \frac{1}{ {x}^(2) } )`
​

Answer 1

If 5x = sec(θ) and 5/x = tan(θ), then x = sec(θ)/5 and 1/x = tan(θ)/5.

Then

5 (x² - 1/x²) = 5 ((sec(θ)/5)² - (tan(θ)/5)²)

… = 5 (sec²(θ)/25 - tan²(θ)/25)

… = sec²(θ)/5 - tan²(θ)/5

Recall that

sec²(θ) = 1 + tan²(θ)

Then

5 (x² - 1/x²) = (sec²(θ) - tan²(θ))/5 = 1/5