Question

If tan \(A = \frac{1}{\sqrt{3}}\) and tan \(B = \sqrt{3}\) then \(\cos A - \cos B - \sin A \cdot \sin B\) will be equal to

• A. 0
• B. \(\frac{1}{2}\)
• C. 1
• D. \(\frac{\sqrt{3}}{2}\)

Hint:

When dealing with trigonometric identities, remember to use the basic trigonometric ratios to simplify the expressions.

Answer 1

The Correct Option is A

We know that \(\tan A = \frac{1}{\sqrt{3}}\), which implies \(\cos A = \frac{1}{2}\) and \(\sin A = \frac{\sqrt{3}}{2}\). Similarly, \(\tan B = \sqrt{3}\), which implies \(\cos B = \frac{1}{2}\) and \(\sin B = \frac{\sqrt{3}}{2}\). Now, calculate: \[ \cos A - \cos B - \sin A \cdot \sin B = \frac{1}{2} - \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = 0 \] Thus, the correct answer is 0.