Question

The value of \(\tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 88^\circ \tan 89^\circ\) will be

• A. 0
• B. $\dfrac{1}{\sqrt{2}}$
• C. $\dfrac{1}{2}$
• D. 1

Hint:

Remember, \(\tan(90^\circ - \theta) = \cot \theta\). Pairing complementary angles often simplifies trigonometric products.

Answer 1

The Correct Option is D

Step 1: Pairing of angles.
Notice that \(\tan(89^\circ) = \cot(1^\circ)\), \(\tan(88^\circ) = \cot(2^\circ)\), and so on.
Step 2: Simplify the product.
\[ \tan 1^\circ \times \tan 89^\circ = \tan 1^\circ \times \cot 1^\circ = 1 \] Similarly, each pair multiplies to 1: \[ (\tan 1^\circ \tan 89^\circ)(\tan 2^\circ \tan 88^\circ) \dots (\tan 44^\circ \tan 46^\circ) \tan 45^\circ = 1 \] Step 3: Conclusion.
Hence, the total product = 1.