Question

Sum of the digits in the value of $\left(1+\tan 5^{\circ}\right)\left(1+\tan 10^{\circ}\right)\left(1+\tan 15^{\circ}\right) \ldots$ $\left(1+\tan 45^{\circ}\right)=$ is______.

Answer 1

$(1+\tan \theta)\left(1+\tan \left(45^{\circ}-\theta\right)\right)$
$=(1+\tan \theta)\left(1+\frac{1-\tan \theta}{1+\tan \theta}\right)$
$=(1+\tan \theta)\left(\frac{2}{1+\tan \theta}\right)=2$
Hence
$LHS =2\left(1+\tan 5^{\circ}\right)\left(1+\tan 40^{\circ}\right)\left(1+\tan 10^{\circ}\right)\left(1+\tan 35^{\circ}\right)$
$\left(1+\tan 15^{\circ}\right)\left(1+\tan 30^{\circ}\right)\left(1+\tan 20^{\circ}\right)\left(1+\tan 25^{\circ}\right)$
$=2 \cdot 2^{4}=2^{5}$