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 Theorem, where the expression is of the form (a+b)n with a=sin(x) b=cos(x) and n=2

2. Apply the formula for the Binomial Theorem, which states (a+b)2=(2/0)a2*b0+(2/1)a1*b1+(2/2)a0*b2)))

3. Substitute the specific values into the expansion:

(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. Evaluate the binomial coefficients (2/0)=1) (2/1)=2) and (2/2)=1)

5. Simplify the expression by performing the multiplication:

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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