Question

Without using trigonometrical tables, find the value of $tan{1}^{\circ }.tan{2}^{\circ }tan{3}^{\circ }\dots tan{89}^{\circ }$ [3 MARKS]

Answer 1

Formula: 1 Mark
Solution: 2 Marks

$tan{1}^{\circ }.tan{2}^{\circ }tan{3}^{\circ }\dots tan{87}^{\circ }tan{88}^{\circ }tan{89}^{\circ }=tan{1}^{\circ }tan{2}^{\circ }tan{3}^{\circ }\dots cot({90}^{\circ }-{87}^{\circ })cot({90}^{\circ }-{88}^{\circ })cot({90}^{\circ }-{89}^{\circ })=tan{1}^{\circ }tan{2}^{\circ }tan{3}^{\circ }\dots cot{3}^{\circ }cot{2}^{\circ }cot{1}^{\circ }=tan{1}^{\circ }tan{2}^{\circ }tan{3}^{\circ }\dots \frac{1}{tan{3}^{\circ }}.\frac{1}{tan{2}^{\circ }}.\frac{1}{tan{1}^{\circ }}=1$