tan1°tan2°tan3°……tan89°=??

Question

tan1°tan2°tan3°……tan89°=??

Answer 1

The answer is 1.

Suppose you have a triangle ABC with angle C being the right angle, Let's call a,b,c the sides opposite angles A, B, and C, respectively. Please draw a diagram for this. Tangent has a weird property. Find the tan A and the tan B. Notice that these are always reciprocals, so when multiplid will give you 1. ( b/a *a/b = 1)

Now let's suppose A=1 degree, That would force B to be 89 degrees. The angles of a triangle add up to 180, but C is 90.

Using the fact above, tan 1 and tan 89 must be reciprocals and multiply to 1.

proceeding we get tan 2 * tan 88 = 1, tan 3 * tan 87 = 1 ...... tan 44  *tan 46 =1. This leaves you with lots of 1's being multiplied by tan 45. But tan 45 =1 by itself.

So your end product for this expression boils down to 1.