Since both cot(θ) and cos(θ) are positive, it follows that sin(θ) is also positive:
cot(θ) = cos(θ)/sin(θ) > 0 ===> sin(θ) > 0
Recall the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
Multiplying both sides by 1/sin²(θ) yields another form of the identity,
sin²(θ)/sin²(θ) + cos²(θ)/sin²(θ) = 1/sin²(θ)
1 + cot²(θ) = csc²(θ)
so that
csc(θ) = + √(1 + cot²(θ))
csc(θ) = √(1 + (3/2)²)
csc(θ) = √13/2
By definition of cosecant, we have
csc(θ) = 1/sin(θ)
so that
sin(θ) = 2/√13
Then using the Pythagorean identity once more, we have
cos(θ) = + √(1 - sin²(θ))
cos(θ) = √(1 - (2/√13)²)
cos(θ) = 3/√13
so that, by definition of secant,
sec(θ) = 1/cos(θ)
sec(θ) = √13/3
