Let I = $\underset{0}{\overset{\mathit{\pi }}{\int }}\frac{{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}{{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+{\mathit{e}}^{-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}$ dx
Using $\underset{0}{\overset{\mathit{a}}{\int }}$ f (x) = $\underset{0}{\overset{\mathit{a}}{\int }}$ f (a - x) dx
I = $\underset{0}{\overset{\mathit{\pi }}{\int }}\frac{{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{\pi }-\mathit{x}\right)}}{{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{\pi }-\mathit{x}\right)}+{\mathit{e}}^{-\mathit{c}\mathit{o}\mathit{s}\left(\mathit{\pi }-\mathit{x}\right)}}$ dx
2I = $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{{\mathit{e}}^{-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}{{\mathit{e}}^{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}+{\mathit{e}}^{-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}}$ dx
I = $\frac{1}{2}\underset{0}{\overset{\mathit{\pi }}{\int }}$ dx = $\frac{1}{2}$ [π - 0] = $\frac{\mathit{\pi }}{2}$
OR
I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ (2 log sin x - log sin 2x) dx
I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ $\left(\log \frac{\mathit{s}\mathit{i}{\mathit{n}}^2\mathit{x}}{2\mathit{s}\mathit{i}\mathit{n}\mathit{x}.\mathit{c}\mathit{o}\mathit{s}\mathit{x}}.\mathit{d}\mathit{x}\right)$
I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log $\left(\frac{\mathit{t}\mathit{a}\mathit{n}\mathit{x}}{2}\right)$ . dx ... (i)
Using property $\underset{0}{\overset{\mathit{a}}{\int }}$ f (x) dx = $\underset{0}{\overset{\mathit{a}}{\int }}$ f (a - x) dx
We get,
I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log $\left(\frac{\mathit{t}\mathit{a}\mathit{n}\left(\frac{\mathit{\pi }}{2}-\mathit{x}\right)}{2}\right)$ dx
⇒ I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log $\left(\frac{\mathit{c}\mathit{o}\mathit{t}\mathit{x}}{2}\right)$ dx ... (ii)
Additing (i)&(ii)
2I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ $\left[\log \left(\frac{\mathit{t}\mathit{a}\mathit{n}\mathit{x}}{2}\right)+\log \left(\frac{\mathit{c}\mathit{o}\mathit{t}\mathit{x}}{2}\right)\right]$ dx
⇒ 2I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log $\left[\left(\frac{\mathit{t}\mathit{a}\mathit{n}\mathit{x}}{2}\right)\left(\frac{\mathit{c}\mathit{o}\mathit{t}\mathit{x}}{2}\right)\right]$ dx
⇒ I = $\frac{1}{2}\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log $\left(\frac{1}{4}\right)$ dx
⇒ I = $\frac{1}{2}$ log $\left(\frac{1}{4}\right)\times \left(\frac{\mathit{\pi }}{2}\right)$
⇒ I = $\frac{1}{2}$ log ${\left(\frac{1}{4}\right)}^{\frac{1}{2}}$ × $\left(\frac{\mathit{\pi }}{2}\right)$
⇒ I = log $\left(\frac{1}{2}\right)\times \left(\frac{\mathit{\pi }}{2}\right)$
⇒ I = $\frac{\mathit{\pi }}{2}\log \frac{1}{2}$