Question:
What is the value of \(\tan 75^\circ + \cot 75^\circ\)?
\(\tan 75^\circ + \cot 75^\circ\) का मान क्या है?
Options:
- 2
- 4
- \(2\sqrt{3}\)
- \(4/\sqrt{3}\)
- 2
- 4
- \(2\sqrt{3}\)
- \(4/\sqrt{3}\)
Correct Answer: b)
Explanation:
Expressing in terms of sine and cosine:
\[\tan 75^\circ + \cot 75^\circ = \frac{\sin 75^\circ}{\cos 75^\circ} + \frac{\cos 75^\circ}{\sin 75^\circ} = \frac{\sin^2 75^\circ + \cos^2 75^\circ}{\sin 75^\circ \cos 75^\circ} = \frac{1}{\sin 75^\circ \cos 75^\circ}\]
Multiplying numerator and denominator by 2:
\[= \frac{2}{2\sin 75^\circ \cos 75^\circ} = \frac{2}{\sin 150^\circ} = \frac{2}{\sin(180^\circ – 30^\circ)} = \frac{2}{\sin 30^\circ} = \frac{2}{1/2} = 4\]
Alternatively: \(\tan 75^\circ = 2 + \sqrt{3}\) and \(\cot 75^\circ = 2 – \sqrt{3}\), so sum is \((2+\sqrt{3}) + (2-\sqrt{3}) = 4\).
Hence, the correct option is (b).
हम जानते हैं कि \(\tan 75^\circ = 2 + \sqrt{3}\) तथा \(\cot 75^\circ = 2 – \sqrt{3}\)।
दोनों का योग:
\[(2 + \sqrt{3}) + (2 – \sqrt{3}) = 4\]
अतः सही विकल्प (b) है।