SOLUTION: Proof Tan 75 - tan 60 = 2 without calculator

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Question 1137387: Proof Tan 75 - tan 60 = 2 without calculator
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Proof :
tan+%2875%29+-+tan%28+60%29+=+2
use: tan+%2875%29=tan%2845%2B30%29 and tan%28+60%29=tan%2830%2B30%29

tan%2845%2B30%29-tan%2830%2B30%29=2

recall: tan%28A+%2B+B%29=+%28tan+A+%2B+tan+B%29+%2F+%281+-tan+A+tan+B%29


since
tan%2845%29=1 and
tan%2830%29=1%2Fsqrt%283%29=sqrt%283%29%2F3
we have






%283+%2Bsqrt%283%29%29%2F+%283+-sqrt%283%29%29+-%282sqrt%283%29%2F3%29+%2F+%282%2F3%29+=2

%283+%2Bsqrt%283%29%29%2F+%283+-sqrt%283%29%29+-%282sqrt%283%29%29+%2F+2+=2

%282%283+%2Bsqrt%283%29%29+-2sqrt%283%29%283+-sqrt%283%29%29%29+%2F+2%283+-sqrt%283%29%29=2

%286+%2B2sqrt%283%29+-6sqrt%283%29+%2B2%2A3%29+%2F+2%283+-sqrt%283%29%29=2

%2812++-4sqrt%283%29+%29+%2F+2%283+-sqrt%283%29%29=2

4%283++-sqrt%283%29+%29+%2F+2%283+-sqrt%283%29%29=2...simplify

4%2F+2=2

2=2