NIMCET 2012 Question Paper with solutions

Let I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log tan x dx ... (i)
Use definite integeral property
I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log tan $\left(\frac{\mathit{\pi }}{2}-\mathit{x}\right)$ dx
= $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log cot x dx ... (ii)
On adding Eqs. (i) and (ii),
2I = $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ (log tan x + log cot x) dx (Since log m + log n = log mn)
= $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log (tan x . cot x) dx
= $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ log 1 dx
= $\underset{0}{\overset{\frac{\mathit{\pi }}{2}}{\int }}$ 0 dx
= 0