Question

Evaluate (sin^2 15 degree + sin^2 75 degree) + sqrt 3 ( tan 13 degree ? tan 60 degree ? tan 27 degree ? cot 20 degree ? cot 70 degree ? tan 77 degree ? tan 63 degree)

Answer 1

Dear student
$$\begin{gathered} \left({\sin }^{2}15°+{\sin }^{2}75°\right)+\sqrt{3}\left(\tan 13°\tan 60°\tan 27°\cot 20°\cot 70°\tan 77°\tan 63°\right) \\ =\left({\sin }^{2}15°+{\sin }^2\left(90°-15°\right)\right)+\sqrt{3}\left[\tan 13°\tan 60°\tan 27°\cot 20°\cot \left(90°-20°\right)\tan \left(90°-13°\right)\tan \left(90°-27°\right)\right] \\ =\left({\sin }^{2}15°+{\cos }^{2}15°\right)+\sqrt{3}\left[\tan 13°\tan 60°\tan 27°\cot 20°\tan 20°\cot 13°\cot 27°\right] \\ =1+\sqrt{3}\left[\tan 60°\right] \\ =1+\sqrt{3}\left(\sqrt{3}\right) \\ =1+3 \\ =4 \end{gathered}$$
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