# Tangent Cosine Sine

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Trigonometry is the branch of Mathematics and it involves the study of the measure of triangles, their angles and their sides. Trigonometry is based on six important trigonometric functions known as sine, cosine, tangent, cosecant, secant and cotangent of a particular angle. Sine, cosine and tangent of an angle are considered the 3 most basic trigonometric functions and the measure of the remaining 3 trigonometric functions can be derived from them. With the help of these trigonometric functions, we can find the measure of the sides and the angles of any given triangle.

Example 1: Express sec(θ)/ cosec(θ) in terms of tangent of the angle, tan(θ).
Given expression: sec(θ)/ cosec(θ)
Now, secant of an angle, sec(θ) can also be written as 1/ cos(θ).
==> sec(θ) = 1/cos(θ).
Similarly, cosec of an angle, cosec(θ) can also be written as 1/sin(θ).
==> cosec(θ) = 1/sin(θ)
Therefore, sec(θ)/ cosec(θ) = (1/cos(θ))/ (1/sin(θ))
Taking the reciprocal, we get: sin(θ)/ cos(θ)
Now we know that, sin(θ)/ cos(θ) = tan(θ).
Therefore, sec(θ)/ cosec(θ) = tan(θ)

Example 2: Simplify the given trigonometric expression [tan(θ) * cot(θ)]/ [cosec(θ)]
Given expression: [tan(θ) * cot(θ)]/ [cosec(θ)]
Now we know that tan(θ) = sin(θ)/ cos(θ)
Also, cot(θ) = 1/tan(θ) and therefore it can also be written as cot(θ) = cos(θ)/ sin(θ).
And, cosec(θ) = 1/sin(θ)
Substituting this, we get: [tan(θ) * cot(θ)]/ [cosec(θ)] = [tan(θ) * 1/tan(θ)]/ [1/sin(θ)]
Hence this gives: 1/ [1/sin(θ)] = sin(θ) (By taking the reciprocal)
Therefore, [tan(θ) * cot(θ)]/ [cosec(θ)] = sin(θ).