Solve for x sec(theta/2)=cos(theta/2)

Problem

sec(θ/2)=cos(θ/2)

Solution

1. Use the reciprocal identity to rewrite the secant function in terms of cosine.

sec(θ/2)=1/cos(θ/2)

2. Substitute the identity back into the original equation.

1/cos(θ/2)=cos(θ/2)

3. Multiply both sides by cos(θ/2) to clear the fraction, noting that cos(θ/2)≠0

1=cos2(θ/2)

4. Take the square root of both sides to solve for the cosine term.

cos(θ/2)=±1

5. Identify the angles where the cosine function equals 1 or −1

θ/2=n*π

6. Solve for θ by multiplying both sides by 2 where n is any integer.

θ=2*n*π

Final Answer

sec(θ/2)=cos(θ/2)⇒θ=2*n*π

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