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

Problem

sec(x/2)=cos(x/2)

Solution

1. Rewrite the secant function in terms of cosine using the reciprocal identity sec(θ)=1/cos(θ)

1/cos(x/2)=cos(x/2)

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

1=cos2(x/2)

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

cos(x/2)=±1

4. Identify the angles where the cosine function equals 1 or −1 which occur at integer multiples of π

x/2=n*π

5. Solve for x by multiplying both sides by 2 where n is any integer.

x=2*n*π

Final Answer

sec(x/2)=cos(x/2)⇒x=2*n*π

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