Standard +0.3 Part (i) is a standard bookwork proof using quotient rule on tan x = sin x/cos x. Part (ii) requires product rule, evaluating at x=π/4, finding tangent equation, and setting x=0—all routine C4 techniques with no novel insight required. Slightly above average due to the algebraic manipulation needed in part (ii).
5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$
The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
5. (i) Use the derivatives of $\sin x$ and $\cos x$ to prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$
The tangent to the curve $y = 2 x \tan x$ at the point where $x = \frac { \pi } { 4 }$ meets the $y$-axis at the point $P$.\\
(ii) Find the $y$-coordinate of $P$ in the form $k \pi ^ { 2 }$ where $k$ is a rational constant.\\
\hfill \mbox{\textit{OCR C4 Q5 [10]}}