Mathematics

Without using trigonometric tables, evaluate the following:
sin² 34° + sin² 56° + 2 tan 18° tan 72° - cot² 30°.

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We need to find the value of:

sin² 34° + sin² 56° + 2 tan 18° tan 72° - cot² 30°

The above equation can be written as,

sin² 34° + sin² (90 - 34)° + 2 tan 18° tan (90 - 18)° - cot² 30°.

As, sin(90 - θ) = cos θ, tan(90 - θ) = cot θ, sin²θ + cos²θ = 1, tan θ.cot θ = 1 and cot 30° = $\sqrt{3}$ . Using in above equation we get,

⇒ sin² 34° + cos² 34° + 2 tan 18° cot 18° - $\sqrt{3}^2$

⇒ 1 + 2 - 3 = 0.

Hence, the value of the above expression is 0.

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