(C) Using the cosine rule,we have $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$,$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$,and $\cos C = \frac{a^2 + b^2 - c^2}{2

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(C) Using the cosine rule,we have $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$,$\cos B = \frac{a^2 + c^2 - b^2}{2ac}$,and $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$.Substituting these into the expression $2(bc \cos A + ca \cos B + ab \cos C)$:$= 2 \left( bc \cdot \frac{b^2 + c^2 - a^2}{2bc} + ca \cdot \frac{a^2 + c^2 - b^2}{2ac} + ab \cdot \frac{a^2 + b^2 - c^2}{2ab} \right)$$= (b^2 + c^2 - a^2) + (a^2 + c^2 - b^2) + (a^2 + b^2 - c^2)$$= a^2 + b^2 + c^2$.

Class 11 Mathematics Trigonometrical Equations Mix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance

Easy

In $\Delta ABC,$ if $2(bc \cos A + ca \cos B + ab \cos C) = $

A
$0$
B
$a + b + c$
C
$a^2 + b^2 + c^2$
D
None of these

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Mix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance

In a triangle,if $r_1 = 2r_2 = 3r_3$,then $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is equal to

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$ABCD$ is a rhombus. The circumradii of $\Delta ABD$ and $\Delta ACD$ are $\frac{25}{2}$ and $25$ respectively. Then the area of the rhombus is .............. $sq. \, unit$.

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In $\triangle ABC$,if $\cos ^2 A + \cos ^2 B + \cos ^2 C = 1$,then $\triangle ABC$ is

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