Question

Determine the value of tan⁡〖45°-tan⁡〖30°〗 〗/(1-tan⁡〖45° .tan⁡〖30°〗 〗 ) .

Answer 1

Question 21163

$$\frac {t a n (4 5 {}^ {\circ} - t a n (3 0 {}^ {\circ}))}{1 - t a n (4 5 {}^ {\circ} - t a n (3 0 {}^ {\circ}))} =$$

Solution.

We first evaluate the expression $\tan(45{}^\circ - \tan(30{}^\circ))$ .

$\tan (45{}^{\circ} - \tan (30{}^{\circ})) = \frac{\tan(45{}^{\circ}) - \tan(\tan(30{}^{\circ}))}{1 + \tan(45{}^{\circ})\tan(30{}^{\circ})} = \frac{1 - \tan(\tan(30{}^{\circ}))}{1 + \tan(\tan(30{}^{\circ}))}$ . Here we use that $\tan (45{}^{\circ}) = 1$ .

Thus,

$$1 - \tan (4 5 {}^ {\circ} - \tan (3 0 {}^ {\circ})) = 1 - \frac {1 - \tan (\tan (3 0 {}^ {\circ}))}{1 + \tan (\tan (3 0 {}^ {\circ}))} = \frac {2 \tan (\tan (3 0 {}^ {\circ}))}{1 + \tan (\tan (3 0 {}^ {\circ}))}.$$

Finally,

$$\frac {t a n (4 5 {}^ {\circ} - t a n (3 0 {}^ {\circ}))}{1 - t a n (4 5 {}^ {\circ} - t a n (3 0 {}^ {\circ}))} = \frac {1 - t a n (t a n (3 0 {}^ {\circ}))}{1 + t a n (t a n (3 0 {}^ {\circ}))} \frac {1 + t a n (t a n (3 0 {}^ {\circ}))}{2 t a n (t a n (3 0 {}^ {\circ}))} = \frac {1 - t a n (t a n (3 0 {}^ {\circ}))}{2 t a n (t a n (3 0 {}^ {\circ}))} = \frac {1 - t a n (\frac {\sqrt {3}}{2})}{2 t a n (\frac {\sqrt {3}}{2})}.$$

Answer. $\frac{1 - \tan\left(\frac{\sqrt{3}}{2}\right)}{2\tan\left(\frac{\sqrt{3}}{2}\right)}$