Question

Evaluate : $\cot 15°\cot 16°\cot 17°\dots .\cot 74°\cot 75°$

Answer 1

Evaluating the given expression:

$\cot 15°\cot 16°\cot 17°\dots .\cot 74°\cot 75°$

$=\cot (90°-75°)\cot (90°-74°)\cot (90°-73°)\dots .cot45°\dots .\cot 74°\cot 75°\left[15°=90°-75°\text{and}cot45°\text{is middle term}\right]$

$=\tan 75°\tan 74°\tan 73°\dots .cot45°\dots .\cot 74°\cot 75°\left[\because cot(90°-A)=\tan A\right]$

$=(\tan 75°\cot 75°)(\tan 74.\cot 74°)\dots .cot45°$

$=\left(\tan 75°\times \frac{1}{\tan 75°}\right)\left(\tan 74°\times \frac{1}{\tan 74°}\right)\dots .cot45°\left[\because cotA=\frac{1}{\tan A}\right]$

$=1\times 1...\times 1$

$=1$
Hence, the value of the given expression is $1$.