Find the derivative of sin(x)cos(x) using the product rule.When I do it I get cos^2(x)-sin^2(x), yet my calculator says it is 2(cos(x))^2 - 1.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative of y = (sin x)(cos x) can be found using the product rule.

y' = [(sin x)(cos x)]'

=> [sin x]'[cos x] + [sin x][cos x]'

=> [cos x][cos x] - [sin x][sin x]

=> (cos x)^2 - (sin x)^2

Use the relation (cos x)^2 + (sin x)^2 = 1 or (sin x)^2 = 1 - (cos x)^2

=> (cos x)^2 - 1 + (cos x)^2

=> 2*(cos x)^2 - 1

The result given by your calculator of 2*(cos x)^2 - 1 is equivalent to the result that you have got of (cos x)^2 - (sin x)^2.

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