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# Proof : Key Stage 5 (A2-Level) : Mathematics Content Knowledge

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Key Stage 5 (A2-Level)
Proof
Question 1 of 1

# 1. How confident are you that you can give a clear and convincing argument to show:

## a. the validity of a mathematical statement?

 Example 1 Examples 2 & 3 Example:Prove that the derivative of tan x is sec2 x. $\tan x = \frac{{\sin x}}{{\cos x}}$ so: $\frac{d}{{dx}}\left( {\tan x} \right) = \frac{d}{{dx}}\left( {\frac{{\sin x}}{{\cos x}}} \right)$ Let u = sin x and v = cos x. Then: $\frac{{du}}{{dx}} = \cos x$  and  $\frac{{dv}}{{dx}} = - \sin x$ Using the quotient rule for differentiation $\frac{d}{{dx}}\left( {\frac{{\sin x}}{{\cos x}}} \right) = \frac{{\cos x\cos x + \sin x\sin x}}{{\cos ^2 x}}$ $= \frac{{\cos ^2 x + \sin ^2 x}}{{\cos ^2 x}} = \frac{1}{{\cos ^2 x}}$ $= \sec ^2 x$