Question

The value of tan 75° – cot75° is

(a) $2\sqrt{3}$

(b) $2+\sqrt{3}$

(c) $2-\sqrt{3}$

(d) 1

Answer 1

$$\begin{gathered} \tan 75°-\cot 75° \\ =\tan 75°-\frac{1}{\tan 75°} \\ =\frac{{\tan }^{2}75°-1}{\tan 75°} \\ =-\frac{\left(1-{\tan }^{2}75°\right)}{\tan 75°}\times \frac{2}{2}\left(\mathrm{multiply}\mathrm{and}\mathrm{divide}\mathrm{by}2\right) \\ =-2\left[\frac{1-{\tan }^{2}75°}{2\tan 75°}\right] \\ =-2\left[\frac{1}{\tan 150°}\right]\left(\mathrm{Using}\mathrm{identity}\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\right) \\ =-2\cot 150° \\ =-2\cot \left(90°+60°\right) \\ =-2\left(-\tan 60°\right)\left(\because \cot \left(90°+\theta \right)=-\tan \theta \right) \\ =2\times \sqrt{3} \\ \therefore \tan 75°-\cot 75°=2\sqrt{3} \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{answer}\mathrm{is}\mathrm{A}. \end{gathered}$$