Given that $y = \ln \left( \sin^2 x \right)$, show that \(\frac{\mathrm{d}y}{\mathrm{d}x

Question

Given that $y = \ln \left( \sin^2 x \right)$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = 2 \cot x$.[2] Hence, show that the value of $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\cot x - \cos^2 2x) \, \mathrm{d}x = a + b \ln 2$, where $a$ and $b$ are constants.[4]

Accepted Answer

$(a)\ \frac{dy}{dx}=2\cot x$ $(b)\ a=-\frac{\pi}{8},\ b=\frac12$

2025 AHS PRELIMS P1 Q12

2025 AHS PRELIMS P1 Q12

Solution:

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