Question

$\tan {75}^{°}-\cot {75}^{°}=$

• A. $2\sqrt{3}$
• B. $2+\sqrt{3}$
• C. $2-\sqrt{3}$
• D. None of these

Answer 1

The correct option is A

$2\sqrt{3}$

Explanation for correct option :

Step-1 : Simplifying the above expression

Given expression is : $\tan 75°-\cot 75°$

Applying trigonometric identities : $\tan A=\frac{\sin A}{\cos A}$ and $cotA=\frac{\cos A}{\sin A}$, we get :

$$\begin{gathered} \tan 75°-\cot 75°=\frac{\sin 75°}{\cos 75°}-\frac{\cos 75°}{\sin 75°} \\ =\frac{{\sin }^{2}75°-{\cos }^{2}75°}{\cos 75°\sin 75°} \end{gathered}$$

Step-2 : Finding the value of ${\sin }^{2}75°-{\cos }^{2}75°$applying the trigonometric identity $\cos 2A={\cos }^{2}A-{\sin }^{2}A$

We have,

${\sin }^{2}75°-{\cos }^{2}75°=-\left({\cos }^{2}75°-{\sin }^{2}75°\right)$

$$\begin{gathered} =-\cos \left(2\times 75°\right) \\ =-\cos \left(150°\right) \\ =-\cos \left(180°-30°\right)\left[\because \cos \left(180°-\theta \right)=-\cos \theta \right] \\ =\cos \left(30°\right) \\ =\frac{\sqrt{3}}{2} \end{gathered}$$

Step-3 : Finding the value of $\cos 75°\sin 75°$ applying the trigonometric identity $\sin 2A=2\sin A\cos A$

We have,

$\cos 75°\sin 75°=\frac{1}{2}\left(2\sin 75°\cos 75°\right)$

$$\begin{gathered} =\frac{1}{2}\sin \left(150°\right) \\ =\frac{1}{2}\sin \left(180°-30°\right) \\ =\frac{1}{2}\sin \left(30°\right)\left[\because \sin \left(180°-\theta \right)=\sin \theta \right] \\ =\frac{1}{2}\times \frac{1}{2} \\ =\frac{1}{4} \end{gathered}$$

Step-4 : Calculating the value of $\tan {75}^{°}-\cot {75}^{°}$

Using the values obtained in Step-2 and Step-3, we get the value of $\tan {75}^{°}-\cot {75}^{°}$ as follows :

$\tan {75}^{°}-\cot {75}^{°}=\frac{{\sin }^{2}75°-{\cos }^{2}75°}{\cos 75°\sin 75°}$

Putting the obtained values, we have

$$\begin{gathered} =\frac{{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{4}}} \\ =2\sqrt{3} \end{gathered}$$

Hence, the correct answer is option (A).