Without using trigonometric tables, prove that:

sin² 20° + sin² 70° - tan² 45° = 0.

To prove,

sin² 20° + sin² 70° - tan² 45° = 0.

Solving, L.H.S. of the equation we get :

sin² 20° + sin² 70° - tan² 45°

We know that,

sin (90 - θ) = cos θ

and

sin² θ + cos² θ = 1.

⇒ sin² 20° + sin² (90° - 20°) - (1)²

⇒ sin² 20° + cos² 20° - 1

⇒ 1 - 1

⇒ 0.

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

Hence, proved that sin² 20° + sin² 70° - tan² 45° = 0.