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Method of Difference

Let be the given series and let

If

Or,

We shall now enlist some of the formulae which help us to express

The above list is by no means exhaustive, but fairly representative meets the needs of the students.

**S**_{n}denote the sum of first**n**terms of the series i.e.**S**_{n}=_{1}+_{2}+ … +_{n}If

_{n}can be expressed in the form**Ø (n + 1) – Ø (n)**, then**S**_{n}= { Ø (2) – Ø (1) + Ø (3) - Ø (2) + …. + { Ø (n + 1) - Ø (n)}Or,

**S**_{n}= Ø (n + 1) - Ø (1)We shall now enlist some of the formulae which help us to express

_{n}in the desired form:**(i) tan = cot – 2 cot 2.**

(ii) cosec 2 = cot – cot 2.

(iii) sec sec (+ ) = cosec (tan ( + ) – tan }).

(iv) tan tan ( + ) = cot (tan ( + ) - (tan ( + ) – tan } – 1.

(v) cosec cosec ( + ) = cosec {cot – cot ( + )}

(vi) tan sec 2 = tan 2 – tan .

(vii) tan 2 tan 2 = tan 2 – 2 tan .

(ix) cos 3 = 4 cos

(x) 4 sin(ii) cosec 2 = cot – cot 2.

(iii) sec sec (+ ) = cosec (tan ( + ) – tan }).

(iv) tan tan ( + ) = cot (tan ( + ) - (tan ( + ) – tan } – 1.

(v) cosec cosec ( + ) = cosec {cot – cot ( + )}

(vi) tan sec 2 = tan 2 – tan .

(vii) tan 2 tan 2 = tan 2 – 2 tan .

(ix) cos 3 = 4 cos

^{3}– 3 cos(x) 4 sin

^{3}= 3 sin – sin 3 .The above list is by no means exhaustive, but fairly representative meets the needs of the students.

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