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Home » Math Homework Help » Algebra Homework Help » Method of Difference
Method of Difference
Let be the given series and let Sn denote the sum of first n terms of the series i.e.

Sn = 1 + 2 + … + n

If n can be expressed in the form Ø (n + 1) – Ø (n), then

Sn = { Ø (2) – Ø (1) + Ø (3) - Ø (2) + …. + { Ø (n + 1) - Ø (n)}

Or, Sn = 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 cos3 – 3 cos
   
(x) 4 sin3 = 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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