Prove the following identity :

Solving L.H.S. of the given equation :

⇒ (sin² θ - 1)(tan² θ + 1) + 1

⇒ (1 - cos² θ - 1).sec² θ + 1

⇒ -cos² θ. sec² θ + 1

⇒ -cos² θ $\times \frac{1}{\text{cos}^2 \text{θ}}$ + 1

⇒ -1 + 1

⇒ 0.

Since, L.H.S. = R.H.S.

Hence, proved that (sin² θ - 1)(tan² θ + 1) + 1 = 0.