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

Problem

sin(x)+cos(x)=1

Solution

1. Square both sides of the equation to eliminate the trigonometric terms' linear relationship, noting that this may introduce extraneous solutions.

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

2. Expand the left side using the binomial expansion formula (a+b)2=a2+2*a*b+b2

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. Isolate the product by subtracting 1 from both sides.

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

5. Apply the double angle identity sin(2*x)=2*sin(x)*cos(x) to simplify the equation further.

sin(2*x)=0

6. Solve for the argument 2*x by identifying where the sine function equals zero.

2*x=n*π

x=(n*π)/2

7. Verify the solutions in the original equation sin(x)+cos(x)=1 to remove extraneous values (like x=π or x=(3*π)/2 where the sum is −1.

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

Want more problems? Check here!