Verify the Identity sin(x+pi/2)=cos(x)

Problem

sin(x+π/2)=cos(x)

Solution

1. Identify the sum formula for the sine function, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

2. Substitute the values A=x and B=π/2 into the sum formula.

3. Apply the formula to the left side of the identity:

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

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

5. Simplify the expression by substituting these values:

sin(x+π/2)=sin(x)*(0)+cos(x)*(1)

6. Finalize the simplification to show that the left side equals the right side:

sin(x+π/2)=cos(x)

Final Answer

sin(x+π/2)=cos(x)

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