Question

Prove the following trigonometric identities:

(i) (1 – sin²θ) sec²θ = 1

(ii) cos²θ (1 + tan²θ) = 1

Answer 1

(i) We have,

`LHS = (1 – sin^2 θ) sec^2 θ`

`= cos^2 θ sec^2 θ [∵ 1 – sin^2 θ = cos^2 θ]`

`=cos ^{2}\theta ( {1}/cos ^{2}\theta)[ \because \ \ \sec \theta =\frac{1}{\cos \theta }]`

= 1 = RHS

(ii) We have,

`LHS = cos^2 θ (1 + tan^2 θ)`

`= cos^2 θ sec^2 θ [∵ 1 + tan^2 θ = sec^2 θ]`

`=\cos ^{2}\theta( 1/\cos^{2}\theta )[\because \ \ \sec \theta =\frac{1}{\cos \theta }]`

= 1 = RHS