Sinnx- r -0 n a r sin r x where n is an odd natural number- thenA- a0-1- a1-nB- a 0-0- a 1- nC- a 0-1- a 1-2 nD- a 0-0- a 1- n

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Principal Solution of Trigonometric Equation

sinnx=∑ r =0 ...

Question

$s i n n x = \sum_{r = 0}^{n} a_{r} s i n^{r} x$ where n is an odd natural number, then

a₀= 1, a₁= 2n

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a₀= 1, a₁= n

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a₀= 0, a₁= n

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a₀= 0, a₁= -n

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Solution

The correct option is C
a₀= 0, a₁= n

sin nx = Im $(e^{i n x})$ = Im $[{(c o s x + i s i n x)}^{n}]$

= $^{n} C_{1} c o s^{n - 1} x . s i n x -^{n} C_{3} c o s^{n - 3} x . s i n^{3} x +^{n} C_{5} c o s^{n - 5} x . s i n^{5} x + . . . . . . . . . . . . . . .$

Since n is odd, let n = 2 $λ$ + 1

$\Rightarrow$ $^{n} C_{1} (c o s^{2} x)^{λ} s i n x -^{n} C_{3} (c o s^{2} x)^{λ - 1} s i n^{3} x + . . . . . . . . . .$

$\Rightarrow$ $^{n} C_{1} (1 - s i n^{2} x)^{λ} s i n x -^{n} C_{3} (1 - s i n^{2} x)^{λ - 1} s i n^{3} x + . . . . . . . . . .$

$\Rightarrow$ $^{n} C_{1} s i n x - (^{n} C_{1} -^{n} C_{1} .^{n} C_{3}) s i n^{3} x + . . . . . . .$

$\Rightarrow$ $a_{0} = 0, a_{1} = n$

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