prove that tan50°tan30°tan20°=tan50°-tan30°-tan20°

Question

this question is from trignometry proving

Answer 1

I think that this problem was misinterpreted by Marlene.  On the left side of the equation we have the product of the tangents of 50, 30 & 20 degrees. On the right side, we have the same tangents & angles, but with the last two subtracted from the first one.

 

I tried this on the calculator, and each side produces the decimal value 0.2504330891. Verily, this is NOT a proof, but it is a demonstration of plausibility, which supports my observation that the right side of the equation includes two subtractions.

 

All of the prior answer's discussion of the four quadrants is not relevant, since all of the angles in this statement lie in quadrant one.

 

Since this request for "proof" only includes three specific angles, and is not a general identity, perhaps this is sufficient in this case.

Answer 2

Hello Sunidhi,

 

To prove this we need to recall the unit circle and the 4 quadrants.

 

Beginning with the top right and moving counter clockwise, we number the quadrants I, II, III, IV.

 

Visually it would look something like this.

 

II   |    I

---------- 

III  |  IV

 

The next thing we need to consider is which trig function are positive and which are negative in each quadrant.

A common pneumonic students use is ASTC, which stands for All Students Take Calculus.  The order of the letters of the pneumonic correspond to the order of numbering the quadrants.  

So in quadrant I, Sin, Cos and Tan are all positive

In quadrant II, only Sin is positive

In quadrant III, only Tan is positive

In quadrant IV, only Cos is positive

 

Now to your question

Tan 50°  Tan 30°  Tan 20° = Tan 50°  -Tan 30°  -Tan 20°

 

All three angles are in the first quadrant, where Tan is positive.

 

So on the left side of the equation we have a positive times a positive times a positive which equals a positive.

And on the right side o the equation we have a positive times a negative times a negative which also equals a positive.