Find :

(a) (sin θ + cosec θ)²

(b) (cos θ + sec θ)²

Using the above results prove the following trigonometry identity :

(sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan² θ + cot² θ

(a) Solving,

⇒ (sin θ + cosec θ)²

⇒ sin² θ + cosec² θ + 2 × sin θ × cosec θ

⇒ sin² θ + 1 + cot² θ + 2 × sin θ × $\dfrac{1}{\text{sin θ}}$

⇒ sin² θ + 1 + cot² θ + 2

⇒ sin² θ + cot² θ + 3.

(b) Solving,

⇒ (cos θ + sec θ)²

⇒ cos² θ + sec² θ + 2 × cos θ × sec θ

⇒ cos² θ + 1 + tan² θ + 2 × cos θ × $\dfrac{1}{\text{cos θ}}$

⇒ cos² θ + tan² θ + 1 + 2

⇒ cos² θ + tan² θ + 3.

Solving,

⇒ (sin θ + cosec θ)² + (cos θ + sec θ)²

⇒ sin² θ + cot² θ + 3 + cos² θ + tan² θ + 3.

⇒ sin² θ + cos² θ + cot² θ + tan² θ + 3 + 3

⇒ 1 + cot² θ + tan² θ + 3 + 3

⇒ 7 + tan² θ + cot² θ.

Hence, proved that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan² θ + cot² θ.