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I have studied that in transmission lines, if λ is the wavelength of the signal and L is the length of the conductor, then at L=λ/4 the transmission line act as an open circuit.

So if we are transmitting 1 GHz signal ( λ = 30cm ) through a conductor of length 7.5 cm, will there not be any output at the end?

Is this the reason why we use waveguides at microwave frequencies?

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You're missing a few criteria for your statement, and that's getting you in trouble. Specifically, you don't say what's at the other end of that λ/4 transmission line, and that's a big part of the answer.

First, let me fill in some detail to your statement to make it fully qualified:

... at L=λ/4 the transmission line acts as an open circuit when the end of the transmission line is shorted (0 Ω).

In this case, you aren't really concerned about there not being any power output at the end of the transmission line because you don't have it connected to anything, it's just a stub. You could definitely use a transmission line that is λ/4 for real power transmission.

The more general statement of your special case is that at λ/4, you have an impedance transformation that can be simplified to the following equation:

\$ \dfrac{Z_{IN}}{Z_0}=\dfrac{Z_0}{Z_{L}}\$

Waveguides are used at microwave frequencies because they have better power handling capability and lower loss than coaxial cable rated for the same frequencies.

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