Given L.H.S. = (sin θ + cosec θ)² + (cos θ + sec θ)²

We know

⇒ sin² θ + cosec² θ + 2 + cos² θ + sec² θ + 2

Also From the Trigonometrical identities

sin²θ + cos²θ = 1

cosec² θ = 1 + cot² θ

sec² θ = 1 + tan² θ

⇒ 1 + 1 + cot² θ + 2 + 1 + tan² θ + 2

⇒ 7 + cot² θ + tan² θ

So, L.H.S = R.H.S

Hence Proved