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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions4 Marks
Question
In any $\triangle A B C$, if $a^2, b^2, c^2$ are in arithmetic progression, then prove that $\cot A,\cot B, \cot C$ are in arithmetic progression.
✓

Answer

Given that $a^2 ,b^2 ,c^2$ are in arithmetic progression.
We need to prove that cot $A$, cot $B$ and cot $C$ are in
arithmetic progression.
$a^2 ,b^2 ,c^2$ are in $A.P$.
$-2 a^2,-2 b^2,-2 c^2$ are in $A.P$
$\left(a^2+b^2+c^2\right)-2 a^2,\left(a^2+b^2+c^2\right)-2 b^2,\left(a^2+b^2+c^2\right)-2 c^2$ are in $A.P$
$\left(b^2+c^2-a^2\right),\left(c^2+a^2-b^2\right),\left(a^2+b^2-c^2\right)$ are in $A.P$
$\frac{b^2+c^2-a^2}{2 a b c}, \frac{c^2+a^2-b^2}{2 a b c}, \frac{a^2+b^2-c^2}{2 a b c}$ are in $A.P$
$\frac{1}{a} \frac{b^2+c^2-a^2}{2 b c}, \frac{1}{b} \frac{c^2+a^2-b^2}{2 a c}, \frac{1}{c} \frac{a^2+b^2-c^2}{2 a b}$ are in $A.P$
$\frac{1}{a} \cos A, \frac{1}{b} \cos B, \frac{1}{c} \cos C$ are in $A.P$
$\frac{k}{a} \cos A, \frac{k}{b} \cos B, \frac{k}{c} \cos C$ are in $A.P$
$\frac{\cos A}{\sin A}, \frac{\cos B}{\sin B}, \frac{\cos C}{\sin C}$ are in $A.P$
$\cot A, \cot B, \cot C$ are in $A.P$

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