Simplify cos(x)-sin(x)

Problem

cos(x)−sin(x)

Solution

1. Identify the expression as a linear combination of sine and cosine in the form a*cos(x)+b*sin(x) where a=1 and b=−1

2. Calculate the amplitude R using the formula R=√(,a2+b2)

R=√(,1+(−1)2)

R=√(,2)

3. Determine the phase shift α such that cos(α)=a/R and sin(α)=−b/R (or using the identity cos(x)−sin(x)=√(,2)*cos(x+π/4).

cos(α)=1/√(,2)

sin(α)=1/√(,2)

α=π/4

4. Apply the harmonic addition theorem to rewrite the expression in the form R*cos(x+α)

cos(x)−sin(x)=√(,2)*cos(x+π/4)

Final Answer

cos(x)−sin(x)=√(,2)*cos(x+π/4)

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