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Transformation Formulae — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTransformation Formulae1 Mark
MCQ
$\text{If }\text{A},\text{B},\text{C}$ are in $\text{A}.\text{P.},$ than $\frac{\sin\text{A}-\sin\text{C}}{\cos\text{C}-\cos\text{A}}=$
  • A
    $\tan\text{B}$
  • ✓
    $\cot\text{B}$
  • C
    $\tan2\text{B}$
  • D
    None of these
✓

Answer

Correct option: B.
$\cot\text{B}$
Since $A,B$ and $C$ are in $A.P,$
$\text{B}-\text{A}=\text{C}-\text{B} $ Or  $,2\text{B}=\text{A}+\text{C}$
$\frac{\sin\text{A}-\sin\text{C}}{\cos\text{C}-\cos\text{A}}$
$=\ \frac{2\sin\Big(\frac{\text{A}-\text{C}}{2}\Big)\cos\Big(\frac{\text{A+C}}{2}\Big)}{-2\sin\Big(\frac{\text{C+A}}{2}\Big)\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)}$
$\Big[\because\ \sin\text{A}-\sin\text{B}=2\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)\cos\Big(\frac{\text{A+B}}{2}\Big)$
$\text{ and }\cos\text{A}-\cos\text{B}=-2\sin\Big(\frac{\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A}-\text{B}}{2}\Big)\Big]$
$=\ \frac{\sin\Big(\frac{\text{A}-\text{C}}{2}\Big)\cos\Big(\frac{\text{A+C}}{2}\Big)}{-\sin\Big(\frac{\text{A+C}}{2}\Big)\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)}$
$=\ \frac{\sin\Big(\frac{\text{A}-\text{C}}{2}\Big)\cos\Big(\frac{\text{A+C}}{2}\Big)}{\sin\Big(\frac{\text{A+C}}{2}\Big)\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)}$
$=\ \frac{\cos\Big(\frac{\text{A+C}}{2}\Big)}{\sin\Big(\frac{\text{A+C}}{2}\Big)}$
$=\ \frac{\cos\text{B}}{\sin\text{B}}$
$=\ \cot\text{B}$

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