# Solve for x sin(x/2)=cos(x/2)

Divide each term in the equation by .
Convert from to .
Replace with an equivalent expression in the numerator.
Remove parentheses.
Rewrite in terms of sines and cosines, then cancel the common factors.
Reorder and .
Rewrite in terms of sines and cosines.
Cancel the common factors.
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
The exact value of is .
Multiply both sides of the equation by .
Simplify both sides of the equation.
Simplify the left side.
Cancel the common factor of .
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
Multiply and .
Divide by .
Simplify .
Cancel the common factor of .
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and .
The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Simplify the expression to find the second solution.
Multiply both sides of the equation by .
Simplify both sides of the equation.
Simplify the left side.
Cancel the common factor of .
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
Multiply and .
Divide by .
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Cancel the common factor of .
Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and .
Subtract from .
The resulting angle of is positive, less than , and coterminal with .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Move to the left of .
The period of the function is so values will repeat every radians in both directions.
, for any integer
Solve for x sin(x/2)=cos(x/2)