Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#7
Question
i. By differentiating , show that if y = cot x then
ii. Hence, show that
By using appropriate trigonometrical identities, find the exact value of
iii.
iv.
Solution
i.
We are given;
Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to
is:
If and
are functions of
, and if
, then;
Let and
, then;
Rule for differentiation of is;
Rule for differentiation of is;
We have the trigonometric identity;
provided that
Therefore, if;
provided that
Hence;
For;
As demonstrated above;
Hence;
ii.
As we have demonstrated in (i) that;
Therefore, the inverse of differentiate ie integral must be as;
Hence;
provided that
iii.
Utilizing the identity;
We can write;
Rule for integration of is:
As demonstrated in (ii);
Hence;
Rule for integration of is:
iv.
We have the trigonometric identity;
Rule for integration of is:
provided that
We have shown in (ii) that;
Therefore;
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