How do you solve Csc^2(a/2)=2 Sec(a)?

1 Answer
Jul 4, 2016

Let us solve the trigonometric equation by using trigonometric identities.

Explanation:

First of all, let us change cosecant and secant for inverses of sine and cosine, respectively:

1/{sin^2 (a/2)} = 2 1/{cos(a)} rightarrow sin^2 (a/2) = 1/2 cos (a) rightarrow
rightarrow 2 sin^2 (a/2) = cos (a)

Now, we shall use the definition of the half-angle sine:

sin (alpha/2) = pm sqrt{{1-cos(alpha)}/2}

So:

sin^2 (alpha/2) = {1-cos(alpha)}/2

And thus, our equation remains:

cancel 2 cdot {1-cos(a)}/{cancel 2} = cos (a) rightarrow 1 - cos(a) = cos(a)

And finally:

1 = 2 cos(a) rightarrow cos(a) = 1/2 rightarrow
rightarrow a = cos^{-1} (1/2) = 60º equiv pi/3 " rad"

Tip: let us check it:

csc^2 (60/2) = 2 cdot sec (60) rightarrow 2^2 = 2 cdot 2 rightarrow 4 = 4