Question

Prove the following trigonometric identities:

(sec  θ + cos θ)(secθ - cos θ) = tan²θ + sin²θ

Answer 1

To prove (sec  θ + cos θ)(secθ - cos θ) = tan²θ + sin²θ

Proof: Use the formula:(a + b) (a - b) = a² - b² on (secθ + cosθ ) (secθ - cosθ) 

Where a = secθ and b = cosθ 

so, 

(secθ + cosθ ) (secθ - cosθ) = sec²θ - cos²θ ...... (1) 

We know,sec²θ = tan²θ + 1

sin²θ + cos²θ = 1 

Use the identities in the eq. (1)(secθ + cosθ ) (secθ - cosθ) = sec²θ - cos²θ 

= (tan²θ + 1) - (1 - sin²θ) 

= tan²θ + 1 - 1 + sin²θ 

= tan²θ + sin²θ 

Hence proved.