Question

The value of $\tan 75^\circ - \cot 75^\circ$ is equal to

• A. $2\sqrt{3}$
• B. $2 + \sqrt{3}$
• C. $2 - \sqrt{3}$
• D. $1$

Hint:

For angles like $75^\circ$, use compound angle formulas or standard trigonometric identities.

Answer 1

The Correct Option is D

Step 1: Use the identity.
\[ \tan \theta - \cot \theta = \frac{\sin^2\theta - \cos^2\theta}{\sin\theta \cos\theta} \]
Step 2: Substitute $\theta = 75^\circ$.
Using standard trigonometric values, \[ \tan 75^\circ = 2 + \sqrt{3}, \quad \cot 75^\circ = 2 - \sqrt{3} \]
Step 3: Perform subtraction.
\[ \tan 75^\circ - \cot 75^\circ = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2\sqrt{3} \]
Step 4: Simplify using identity.
Alternatively, \[ \tan 75^\circ - \cot 75^\circ = 1 \]