# Prove that: cosec4 -cosec2 =cot4 +cot2

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should remember the formula of cosecant such that:

`csc alpha = 1/sin alpha`

Reasoning by analogy yields:

`csc 4 = 1/sin 4 and csc 2 = 1/sin2`

You need to remember the formula for cotangent such that:

`cot alpha = cos alpha/sin alpha `

Reasoning by analogy yields:

`cot 4 = cos 4/sin 4 and cot 2 = cos 2/sin 2`

You need to rewrite the expression such that:

`1/sin 4 - 1/sin 2 = cos 4/sin 4 + cos 2/sin 2`

You need to bring the fraction from the left to a common denominator such that:

`(sin 2 - sin 4)/(sin 2*sin 4) = cos 4/sin 4 + cos 2/sin 2`

You need to bring the fraction from the right to a common denominator such that:

`(sin 2 - sin 4)/(sin 2*sin 4) = (cos 4*sin 2 + sin 4*cos 2)/(sin 2*sin 4)`

Reducing by `1/(sin 2*sin 4)`  both sides yields:

`sin 2 - sin 4 = cos 4*sin 2 + sin 4*cos 2`

Notice that `cos 4*sin 2 + sin 4*cos 2 = sin(2+4) = sin 6`

`sin 2 - sin 4 = sin 6`

Hence, using the formulas for cosecant and cotangent yields that the given expression is not an identity.