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Question

prove that 1 + sec 20 = cot 40 cot 30


Solution

To start with, cot(70) = cot(40 + 30) = (cot(40)*cot(30) - 1) / (cot(40) + cot(30)), so 

cot(40)*cot(30) = cot(70)*(cot(40) + cot(30)) + 1. 

But cot(70) = tan(20) = sin(20)*sec(20), so if we can show that 

sin(20) * (cot(40) + cot(30)) = 1, then the proof will be complete. 

So, cot(40) + cot(30) = cos(40)/sin(40) + cos(30)/sin(30) = 

(cos(40)sin(30) + sin(40)cos(30)) / (sin(40)*sin(30)) = 

sin(70) / ((1/2)*sin(40)) = cos(20) / (1/2)*2*sin(20)*cos(20)) = 1 / sin(20), 

and so sin(20) * (cot(40) + cot(30)) = 1 as desired.

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