What is antiderivative of y = (cosx)^4-(sinx)^4)/((cosx)^2-(sinx)^2)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the anti-derivative of the given function, hence, you need to find its indefinite integral, such that:

`int (cos^4 x - sin^4 x)/(cos^2 x - sin^2 x)dx`

You need to convert the difference of squares to numerator into a product, such that:

`int ((cos^2 x - sin^2 x)(cos^2 x + sin^2 x))/(cos^2 x - sin^2 x)dx`

You need to reduce the duplicate factors, such that:

`int (cos^2 x + sin^2 x) dx`

Using the basic formula of trigonometry, `cos^2 x + sin^2 x = 1` , yields:

`int (cos^2 x + sin^2 x) dx = int 1*dx = x + c`

Hence, evaluating the anti-derivative of the given function, using the special products and the basic formula of trigonometry, yields `int (cos^4 x - sin^4 x)/(cos^2 x - sin^2 x)dx = x + c.`

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