Express (ctgx)^2 - (cosx)^2 as a single fraction in terms of sinx, cosx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to replace `cos x/sin x` for the function `cot x` , such that:

`(cot x)^2 - (cos x)^2 = (cos x/sin x)^2 - (cos x)^2`

Bringing the terms to a common denominator yields:

`(cot x)^2 - (cos x)^2 = (cos^2 x - sin^2 x*cos^2 x)/(sin^2 x)`

You may factor out `cos^2 x` to numerator, such that:

`(cot x)^2 - (cos x)^2 = (cos^2 x(1 - sin^2 x))/(sin^2 x)`

Using the Pythagorean trigonometric identity, yields:

`1 - sin^2 x = cos^2 x`

`(cot x)^2 - (cos x)^2 = (cos^2 x*cos^2 x)/(sin^2 x)`

`(cot x)^2 - (cos x)^2 = (cos^4 x)/(sin^2 x)`

Hence, expressing the difference of squares as a single fraction, in terms of `sin x` and `cos x` , yields `(cot x)^2 - (cos x)^2 = (cos^4 x)/(sin^2 x).`

(cot x)^2 - (cos x)^2 = (cos^4 x)/(sin^2 x)


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