Solve for x cos(x+pi/4)-cos(x-pi/4)=1

Problem

cos(x+π/4)−cos(x−π/4)=1

Solution

1. Apply the sum-to-product formula for the difference of two cosines, which states cos(A)−cos(B)=−2*sin((A+B)/2)*sin((A−B)/2)

2. Identify the terms where A=x+π/4 and B=x−π/4

3. Calculate the arguments for the sine functions: (A+B)/2=(2*x)/2=x and (A−B)/2=(π/2)/2=π/4

4. Substitute the values back into the formula to rewrite the equation.

−2*sin(x)*sin(π/4)=1

5. Evaluate the constant sin(π/4)=√(,2)/2

−2*sin(x)*(√(,2)/2)=1

6. Simplify the expression to isolate the sine term.

−√(,2)*sin(x)=1

7. Solve for sin(x) by dividing both sides by −√(,2)

sin(x)=−1/√(,2)

8. Rationalize the denominator to get the standard value.

sin(x)=−√(,2)/2

9. Find the general solution for x where the sine is −√(,2)/2 which occurs in the third and fourth quadrants.

x=(5*π)/4+2*k*π

x=(7*π)/4+2*k*π

Final Answer

cos(x+π/4)−cos(x−π/4)=1⇒x=(5*π)/4+2*k*π,(7*π)/4+2*k*π

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