Question

If cos A = `1/sqrt2` and sin B = `sqrt3/2` the value of `(tan B-tan A)/(1+ tan A tan B)` is ______.

Options

• `2 - sqrt3`

• `2 + sqrt3`

• `sqrt3 - 2`

• `sqrt3 + 2`

Answer 1

If cos A = `1/sqrt2` and sin B = `sqrt3/2` the value of `(tan B-tan A)/(1+ tan A tan B)` is `bbunderline(2 - sqrt3)`.

Explanation:

Given \[\cos A = \frac{1}{\sqrt{2}}\], the angle A = 45°, which gives tan A = 1. Given \[\sin B = \frac{\sqrt{3}}{2}\], the angle B = 60°, which gives \[\tan B = \sqrt{3}\]. Substituting these values into the expression:

\[\frac{\tan B - \tan A}{1 + \tan A \tan B} = \frac{\sqrt{3} - 1}{1 + 1 \times \sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\]

Rationalizing the denominator:

\[\frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{3 + 1 - 2\sqrt{3}}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3}\]