Question

evalute
tan 15 tan 25 tan 60 tan 65 tan 75

Answer 1

Recall the identity $\tan(90-\theta)=\cot(\theta)$ noticing that $65=90-25$ and $75=90-15$ :

$\tan(15)\tan(25)\tan(60)\tan(65)\tan(75)\\=\tan(15)\tan(25)\tan(60)\tan(90-25)\tan(90-15)\\=\tan(15)\tan(25)\tan(60)\cot(25)\cot(15)\\=\tan(15)\cot(15)\tan(25)\cot(25)\tan(60)\\=\tan(60)\\=\sqrt{3}$

Answer 2

$tan15tan25tan60tan65tan75$

=$tan(90-75)tan(90-65)tan60tan65tan75$

=$cot 75cot65tan60tan65tan75$

cot75-tan75 and cot 65-tan65 both cancelled

=$tan60$

=$\sqrt{3}$
                                                                 hope you understood