Prove that $$ sec^2 x - cosec^2 x = tan^2 x - cot^2 x $$ (ii) Given that y = arccos x, -1 \leq x \leq 1 and 0 \leq y \leq \pi, a) express arcsin x in terms of y - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 6

Since $y = arccos x$, we can express x in terms of y:

$x = cos(y).$

Now, using the Pythagorean identity: $sin^2 y + cos^2 y = 1$, we find:$

$sin^2 y = 1 - x^2$ Thus: $arcsin x = \frac{\pi}{2} - y.$

From the previous step we derived that $arcsin x = \frac{\pi}{2} - y.$ Thus, substituting into the expression:

$arcsin x + arcsin x = 2\left(\frac{\pi}{2} - y\right) = \pi - 2y.$

Substituting back for y:

$= \pi - 2arccos x.$