Question

3cot 31° tan 15° cot 27° tan 75° cot 63° cot 59°

Answer 1

$$\begin{gathered} 3\cot 31°\tan 15°\cot 27°\tan 75°\cot 63°\cot 59° \\ =3\cot \left(90°-59°\right)\tan 15°\cot \left(90°-63°\right)\tan 75°\cot 63°\cot 59° \\ =3\tan 59°\tan 15°\tan 63°\tan 75°\cot 63°\cot 59°\left(\because \cot \left(90°-\theta \right)=\tan \theta \right) \\ =3\tan 59°\tan \left(90°-75°\right)\tan 63°\tan 75°\cot 63°\cot 59° \\ =3\tan 59°\cot 75°\tan 63°\tan 75°\cot 63°\cot 59°\left(\because \tan \left(90°-\theta \right)=\cot \theta \right) \\ =3\left(\tan 59°\cot 59°\right)\left(\tan 63°\cot 63°\right)\left(\cot 75°\tan 75°\right) \\ =3\left(\tan 59°\frac{1}{\tan 59°}\right)\left(\tan 63°\frac{1}{\tan 63°}\right)\left(\frac{1}{\tan 75°}\tan 75°\right)\left(\because \cot \theta =\frac{1}{\tan \theta }\right) \\ =3 \\ \mathrm{Hence},3\cot 31°\tan 15°\cot 27°\tan 75°\cot 63°\cot 59°=3. \end{gathered}$$