$=\sin3\text{x}+(\sin2\text{x}-\sin\text{x})$
$=\sin3\text{x}+\Big[2\cos\Big(\frac{2\text{x}+\text{x}}{2}\Big)\sin\Big(\frac{2\text{x}-\text{x}}{2}\Big)\Big]$$\Big[\sin\text{A}-\sin\text{B}=2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big)\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big]$
$=\sin3\text{x}+\Big[2\cos\Big(\frac{3\text{x}}{2}\Big)+\sin\Big(\frac{\text{x}}{2}\Big)\Big]$
$=\sin3\text{x}+2\cos\frac{3\text{x}}{2}\sin\frac{\text{x}}{2}$
$=2\sin\frac{3\text{x}}{2}.\cos\frac{3\text{x}}{2}+2\cos\frac{3\text{x}}{2}\sin\frac{\text{x}}{2}$ $[\sin2\text{A}=2\sin\text{A}.\cos\text{B}]$
$=2\sin\cos\Big(\frac{3\text{x}}{2}\Big)\Big[\sin\Big(\frac{3\text{x}}{2}\Big)+\sin\Big(\frac{\text{x}}{2}\Big)\Big]$
$=2\cos\Big(\frac{3\text{x}}{2}\Big)\begin{bmatrix}2\sin\begin{Bmatrix}\frac{\Big(\frac{3\text{x}}{2}\Big)+\Big(\frac{\text{x}}{2}\Big)}{2}\end{Bmatrix}\end{bmatrix}\cos\begin{Bmatrix}\frac{\Big(\frac{3\text{x}}{2}\Big)-\Big(\frac{\text{x}}{2}\Big)}{2}\end{Bmatrix}$ $\Big[\sin\text{A}+\sin\text{B}=2\sin\Big(\frac{\text{A}+\text{B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big]$
$=2\cos\Big(\frac{3\text{x}}{2}\Big).2\sin\text{x}\cos\Big(\frac{\text{x}}{2}\Big)$
$=4\sin\text{x}\cos\Big(\frac{\text{x}}{2}\Big)\cos\Big(\frac{3\text{x}}{2}\Big)=\text{R.H.S.}$
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| column I | column II | ||
| (a) | Throught the point (2, 1) is | (a) | 2x - y = 4 |
| (b) | perpendicular to the line x + 2y + 1 = 0 is | (b) | x + y - 5 = 0 |
| (c) | parpallel to the line 3x + 4y + 5 = 0 | (c) | x - y - 1 |
| (d) | Equally inlined to the axis is | (d) | 3x - 4y - 1 = 0 |
Find a, if the coefficients of x2 and x3 in the expansion of $(3+\text{ax})^{9}$ are equal.