Question

Prove each of the following identities:
1. (cosec2θ – 1) sin2θ = cos2θ

2. sin4A + sin2A cos2A = sin2A

3. cos2θ (1 + tan2θ) = 1

4. (1 + tan2θ) sin2θ = tan2θ.

Answer 1

Q 1.

$( {cosec}^{2} \theta - 1) {sin}^{2} \theta \\ = {cot}^{2} \theta \: {sin}^{2} \theta \\ = \frac{ {cos}^{2} \theta }{ {sin}^{2} \theta } {sin}^{2} \theta \\ = {cos}^{2} \theta$

Q 2.

${sin}^{4} a + {sin}^{2} a {cos}^{2} a \\ = {sin}^{2} a( {sin}^{2} a + {cos}^{2} a) \\ = {sin}^{2} a$

Q 3.

${cos}^{2} \theta(1 + {tan}^{2} \theta) \\ = {cos}^{2} \theta {sec}^{2} \theta \\ = 1$

Q 4.

$(1 + {tan}^{2} \theta) {sin}^{2} \theta \\ = {sec}^{2} \theta \: {sin}^{2} \theta \\ = \frac{ {sin}^{2} \theta}{ {cos}^{2} \theta } \\ = {tan}^{2} \theta$