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TRIGONOMETRIC FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTRIGONOMETRIC FUNCTIONS1 Mark
MCQ
If $\sin(\text{B+C}-\text{A}),\sin(\text{C+A}-\text{B}),\sin(\text{A+B}-\text{C})$ are in A.P., than $\cot\text{A},\cot\text{B},\cot\text{C}$ are in
  • A
    $\text{GP}$
  • ✓
    $\text{HP}$
  • C
    $\text{AP}$
  • D
    None of these
✓

Answer

Correct option: B.
$\text{HP}$
Given:
$\sin(\text{B+C}-\text{A}),\sin(\text{C+A}-\text{B}),\text{ and }\sin(\text{A+B}-\text{C})$ are in A.P.
$\Rightarrow\ \sin(\text{C+A}-\text{B})-\sin(\text{B+C}-\text{A})\\ \ \ \ =\sin(\text{A+B}-\text{C})-\sin(\text{C+A}-\text{B})$
$\Rightarrow\ 2\sin\Big(\frac{\text{C+A}-\text{B}-\text{B}-\text{C+A}}{2}\Big)\cos\Big(\frac{\text{C+A}-\text{B+B+C}-\text{A}}{2}\Big)\\ \ \ \ =2\sin\Big(\frac{\text{A+B}-\text{C}-\text{C}-\text{A+B}}{2}\Big)\cos\Big(\frac{\text{A+B}-\text{C+C+A}-\text{B}}{2}\Big)$
$\Rightarrow\ \sin(\text{A}-\text{B})\cos\text{C}=\sin(\text{B}-\text{C})\cos\text{A}$
$\Rightarrow\ \sin\text{A}\cos\text{B}\cos\text{C}-\cos\text{A}\sin\text{B}\cos\text{C}\\ \ \ =\sin\text{B}\cos\text{C}\cos\text{A}-\cos\text{B}\sin\text{C}\cos\text{A}$
$\Rightarrow\ 2\sin\text{B}\cos\text{A}\cos\text{C}=\sin\text{A}\cos\text{B}\cos\text{C}+\cos\text{A}\cos\text{B}\sin\text{C}$
Dividing both sides by $\cos\text{A}\cos\text{B}\cos\text{C}:$
$2\tan\text{B}=\tan\text{A}+\tan\text{C}$
$\Rightarrow\ \frac{2}{\cot\text{B}}=\frac{1}{\cot\text{A}}+\frac{1}{\cot\text{C}}$
Hence, $\cot\text{A}, \cot\text{B}\text{ and }\cot\text{C}$ are in H.P.

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