Question #217345

Assume f(x) = sin(x+/π4)cos(π/4) + cos(x+/π4)sin(π/4) for all x elements of R. Which of the following statements is correct?

A. f(x) = -cos(x)

B. f(x) = sin(x)

C. f(x) = cos(x)

D. f(x) = -sin(x)


1
Expert's answer
2021-08-06T10:22:25-0400

Solution:

f(x)=sin(x+π4)cos(π4)+cos(x+π4)sin(π4)f(x) = \sin(x+\fracπ4)\cos(\fracπ4) + \cos(x+\fracπ4)\sin(\fracπ4)

Using sinAcosB+cosAsinB=sin(A+B)\sin A \cos B +\cos A \sin B=\sin (A+B)

f(x)=sin(x+π4+π4)=sin(π2+x)=cosxf(x) = \sin(x+\fracπ4+\fracπ4)=\sin(\fracπ2+x)=\cos x

Hence, option (C) is correct.


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