Solve for x sin(x+pi/2)=cos(x)

Problem

sin(x+π/2)=cos(x)

Solution

1. Apply the sum identity for sine, which states sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

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

sin(x)*cos(π/2)+cos(x)*sin(π/2)=cos(x)

3. Evaluate the trigonometric constants cos(π/2)=0 and sin(π/2)=1

sin(x)⋅0+cos(x)⋅1=cos(x)

4. Simplify the left side of the equation.

cos(x)=cos(x)

5. Identify that the equation is an identity, meaning it is true for all values of x where the functions are defined.

Final Answer

sin(x+π/2)=cos(x)⇒x∈ℝ

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