Simplify cos(x-pi/4)

Problem

cos(x−π/4)

Solution

1. Identify the appropriate trigonometric identity for the cosine of a difference, which is cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

2. Substitute the values A=x and B=π/4 into the identity.

cos(x−π/4)=cos(x)*cos(π/4)+sin(x)*sin(π/4)

3. Evaluate the trigonometric constants cos(π/4) and sin(π/4) both of which equal √(,2)/2

cos(x−π/4)=cos(x)*(√(,2)/2)+sin(x)*(√(,2)/2)

4. Factor out the common term √(,2)/2 to simplify the expression.

cos(x−π/4)=√(,2)/2*(cos(x)+sin(x))

Final Answer

cos(x−π/4)=√(,2)/2*(cos(x)+sin(x))

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