Question

Show that:
cot40°×cot25°-cot65°×cot40°=2
​

Answer 1

Answer:

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.

Step-by-step explanation:

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