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)
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)f(x)=sin(x+4π)cos(4π)+cos(x+4π)sin(4π)
Using sinAcosB+cosAsinB=sin(A+B)\sin A \cos B +\cos A \sin B=\sin (A+B)sinAcosB+cosAsinB=sin(A+B)
f(x)=sin(x+π4+π4)=sin(π2+x)=cosxf(x) = \sin(x+\fracπ4+\fracπ4)=\sin(\fracπ2+x)=\cos xf(x)=sin(x+4π+4π)=sin(2π+x)=cosx
Hence, option (C) is correct.
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