Solve for x sin(x)+cos(x)=1

Problem

sin(x)+cos(x)=1

Solution

1. Square both sides of the equation to relate the trigonometric functions, noting that this may introduce extraneous solutions.

sin(x)+cos(x)=1

(sin(x)+cos(x))2=1

2. Expand the left side using the binomial expansion formula.

sin2(x)+2*sin(x)*cos(x)+cos2(x)=1

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the expression.

1+2*sin(x)*cos(x)=1

4. Subtract 1 from both sides to isolate the product term.

2*sin(x)*cos(x)=0

5. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x)

sin(2*x)=0

6. Solve for the general angle by identifying where the sine function equals zero.

2*x=n*π

x=(n*π)/2

7. Verify solutions in the original equation sin(x)+cos(x)=1 to exclude extraneous values.

If *n=0,x=0⇒sin(0)+cos(0)=0+1=1

If *n=1,x=π/2⇒sin(π/2)+cos(π/2)=1+0=1

If *n=2,x=π⇒sin(π)+cos(π)=0−1=−1

If *n=3,x=(3*π)/2⇒sin((3*π)/2)+cos((3*π)/2)=−1+0=−1

8. Generalize the valid solutions based on the verification.

x=2*n*π

x=2*n*π+π/2

Final Answer

sin(x)+cos(x)=1⇒x=2*n*π,2*n*π+π/2

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