Expand Using the Binomial Theorem (sin(x)-cos(x))^2

Problem

(sin(x)−cos(x))2

Solution

1. Identify the terms and the exponent for the binomial expansion (a+b)n where a=sin(x) b=−cos(x) and n=2

2. Apply the Binomial Theorem formula, which states (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

3. Write the expansion by calculating each term for k=0,1,2

(2/0)(sin(x))2*(−cos(x))0+(2/1)(sin(x))1*(−cos(x))1+(2/2)(sin(x))0*(−cos(x))2)))

4. Simplify the binomial coefficients and the signs of each term.

1⋅sin2(x)⋅1+2⋅sin(x)⋅(−cos(x))+1⋅1⋅cos2(x)

5. Combine the terms into the final expanded expression.

sin2(x)−2*sin(x)*cos(x)+cos2(x)

Final Answer

(sin(x)−cos(x))2=sin2(x)−2*sin(x)*cos(x)+cos2(x)

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