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QuestionSimplify the expression (sin⁡θ−cos⁡θ)2(\sin \theta - \cos \theta)^{2}.

Studdy Solution

STEP 1

Assumptions1. We are given the expression (sin⁡θ−cos⁡θ)(\sin \theta-\cos \theta)^{}. . We need to simplify this expression.

STEP 2

We can simplify this expression by expanding the square. The square of a binomial (a-b)^2 is given by a^2 -2ab + b^2.
(sin⁡θ−cos⁡θ)2=(sin⁡θ)2−2sin⁡θcos⁡θ+(cos⁡θ)2(\sin \theta-\cos \theta)^{2} = (\sin \theta)^{2} -2 \sin \theta \cos \theta + (\cos \theta)^{2}

STEP 3

We know from the Pythagorean identity in trigonometry that (sin⁡θ)2+(cos⁡θ)2=1(\sin \theta)^{2} + (\cos \theta)^{2} =1. We can substitute this into our expression.
(sin⁡θ−cos⁡θ)2=1−2sin⁡θcos⁡θ(\sin \theta-\cos \theta)^{2} =1 -2 \sin \theta \cos \thetaSo, the simplified form of (sin⁡θ−cos⁡θ)2(\sin \theta-\cos \theta)^{2} is 1−2sin⁡θcos⁡θ1 -2 \sin \theta \cos \theta.

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