A geometry problem by Anoopam Mishra

Geometry Level pending

Let $a + b + c = 180$ and $\cot a + \cot b + \cot c = \cot x$ . Then
$\csc^2 a + \csc^2 b + \csc^2 c$ is equal to

(cos x)^2 (cosec x)^2 (sec x)^2 (sin x)^2

1 solution

Anoopam Mishra
Sep 4, 2014

Squaring both the sides we get
(cot a)^2 + (cot b)^2 + (cot c)^2 + 2(cotacotb + cotbcotc + cotccota) = (cot x)^2

cotacotb + cotbcotc + cotccota = 1 because a+b+c = 180

Now adding 1 to both the sides we get (cot a)^2 + (cot b)^2 + (cot c)^2 + 3 = (cot x)^2 + 1

As we know that (cot m)^2 + 1 = (cosec m)^2
The equation becomes
(cosec a)^2 + (cosec b)^2 + (cosec c)^2 = (cosec x)^2