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Sum of n Terms

In Δ ABC a...

Question

In $Δ A B C$
$a^{2}, b^{2}, c^{2}$ are in A.P.
Show $cot A, cot B, cot C$ are in A.P

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Solution

Given $a^{2}, b^{2}, c^{2}$ or in $A . P$
$\Rightarrow b^{2} - a^{2} = c^{2} - b^{2}$
$\Rightarrow K^{2} {sin}^{2} B - K^{2} {sin}^{2} A = K^{2} {sin}^{2} C - K^{2} {sin}^{2} B$
$\Rightarrow sin (B + A) . sin (B - A) = sin (C + B) . sin (C - B)$
$\Rightarrow sin (π - C) . sin (B - A)$
$= sin (π - A) sin (C - B)$
$\Rightarrow sin C . sin (B - A) = sin A . sin (C - B)$
$\Rightarrow \frac{sin (B - A)}{sin A \times sin B} = \frac{sin (C - B)}{sin C \times sin B}$
$\Rightarrow \frac{sin B cos A}{sin A sin B} - \frac{cos B sin A}{sin A . sin B} = \frac{sin C cos B}{sin C . sin B} - \frac{cos C . sin B}{sin C . sin B}$
$\Rightarrow cot A - cot B = cot B - cot C$
$\Rightarrow cot A, cot B, cot C$ or in $A P$
$\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c} = K$
$A + B + C = π$

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$I n a n y Δ A B C, i f a^{2}, b^{2}, c^{2} are in A.P., prove that cot A, cot B and cot C are also in A.P.$

Q. In ∆ABC, if a², b² and c² are in A.P., prove that cot A, cot B and cot C are also in A.P.

Δ A B C

a^{2}, b^{2}, c^{2}

are in AP then prove that

cot A, cot B, cot C

If a, b, c are in A.P., then (b+c)²-a², (a+c)² -b², (a+b)² -c² are in....

a^{2}, b^{2}, c^{2}

are in A.P., then

cot A, cot B, cot C

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