(a) Given that $$\frac{d}{dx}(\cos x) = -\sin x$$ show that $$\frac{d}{dx}(\sec x) = \sec x \tan x.$$ Given that $$x = \sec 2y$$ (b) find \( \frac{dx}{dy} \) in terms of \( y \) - Edexcel - A-Level Maths Pure - Question 2 - 2010 - Paper 5

Given that $x = \sec 2y,$ we differentiate with respect to ( y ):

$\frac{dx}{dy} = \frac{d}{dy}(\sec 2y) = 2 \sec 2y \tan 2y.$

Thus, we have:

$\frac{dx}{dy} = 2\sec 2y \tan 2y.$

From the previous step, we have:

$\frac{dx}{dy} = 2\sec 2y \tan 2y.$

To find ( \frac{dy}{dx} ), we take the reciprocal:

$\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{2\sec 2y \tan 2y}.$

Thus,

$\frac{dy}{dx} = \frac{1}{2\sec 2y \tan 2y}.$