Question

Proof : tan10 x tan20 x tan…70 x tan80=1

Answer 1

This follows from the identity

$\cot(x) = \dfrac1{\tan(x)} = \tan(90^\circ - x)$

The overall product is 1 because

$\tan(10^\circ) \tan(80^\circ) = \cot(80^\circ) \tan(80^\circ) = 1$

$\tan(20^\circ) \tan(70^\circ) = \cot(70^\circ) \tan(70^\circ) = 1$

$\tan(30^\circ) \tan(60^\circ) = \cot(60^\circ) \tan(60^\circ) = 1$

$\tan(40^\circ) \tan(50^\circ) = \cot(50^\circ) \tan(50^\circ) = 1$