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.