prove the following identity algebraically. cos angle(sec angle-csc angle)= 1-cot angleangle means theta.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Use the following notation angle = `alpha` .

You need to prove the identity `cos alpha*(sec alpha - csc alpha) = 1 - cot alpha`

You should replace `sec alpha`  by `1/cos alpha`  and `csc alpha`  by `1/sin alpha` .

The left side becomes: `cos alpha*(1/cos alpha - 1/sin alpha).`

Bringing the terms in the brackets to a common denominator yields:

`(cos alpha*(sin alpha - cos alpha))/(cos alpha*sin alpha).`

Reducing by `cos alpha`  yields:

`(sin alpha - cos alpha)/sin alpha`

Opening the brackets yields: `sin alpha/sin alpha - cos alpha/sin alpha`

The ratio `cos alpha/sin alpha`  denotes the cotangent function, therefore `sin alpha/sin alpha - cos alpha/sin alpha = 1 - cot alpha.`

This final result proves that the identity is checked.