(B) Given that $A, B, C$ are angles of a triangle,$A + B + C = \pi$,so $C = \pi - (A + B)$.We know that $\sin^2 A + \sin^2 B + \sin^2 C = 2 + 2\cos A

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(B) Given that $A, B, C$ are angles of a triangle,$A + B + C = \pi$,so $C = \pi - (A + B)$.We know that $\sin^2 A + \sin^2 B + \sin^2 C = 2 + 2\cos A \cos B \cos C$.Starting with the expression: $\sin^2 A + \sin^2 B + \sin^2 C - 2\cos A \cos B \cos C$.Using the identity $\sin^2 A + \sin^2 B + \sin^2 C = 2 + 2\cos A \cos B \cos C$,we substitute this into the expression:$= (2 + 2\cos A \cos B \cos C) - 2\cos A \cos B \cos C$$= 2$.

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

Medium

If $A, B, C$ are the angles of a triangle,then $\sin^2 A + \sin^2 B + \sin^2 C - 2\cos A \cos B \cos C = $

A
$1$
B
$2$
C
$3$
D
$4$

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Class 11

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Trigonometrical Equations

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

The angles $A, B$ and $C$ of a triangle $ABC$ are in $A.P.$ and $a : b = 1 : \sqrt{3}$. If $c = 4 \text{ cm}$,then the area (in $\text{sq. cm}$) of this triangle is:

DifficultJEE MAIN 2019

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In a triangle $ABC$,the sides $a, b, c$ are the roots of the equation $x^3-11x^2+38x-40=0$. Then,find the value of $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$.

MediumMHT CET 2025

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The number of solutions of the equation $\left( 1 - \frac{1}{2 \sin x} \right) \cos^2 2x = 2 \sin x - 3 + \frac{1}{\sin x}$ in the interval $[0, 4\pi]$ is:

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In $\triangle ABC$,if $\cos 3A + \cos 3B + \cos 3C + \cos 3\pi = 0$,then the least value of the sum of two of its angles is

DifficultAP EAMCET 2018

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In a $\triangle ABC$,with usual notation,match the items in List-$I$ with the items in List-$II$ and choose the correct option.
List-$I$ List-$II$
$(A) \ r_1 r_2 \sqrt{\frac{4R-r_1-r_2}{r_1+r_2}}$ $1. \ b$
$(B) \ \frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}$ $2. \ a^2, b^2, c^2 \text{ are in } AP$
$(C) \ \frac{a}{c} = \frac{\sin(A-B)}{\sin(B-C)}$ $3. \ \Delta$
$(D) \ bc \cos^2 \frac{A}{2}$ $4. \ R r_1 r_2 r_3$
$5. \ s(s-a)$

DifficultAP EAMCET 2019

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List-$I$	List-$II$
$(A) \ r_1 r_2 \sqrt{\frac{4R-r_1-r_2}{r_1+r_2}}$	$1. \ b$
$(B) \ \frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}$	$2. \ a^2, b^2, c^2 \text{ are in } AP$
$(C) \ \frac{a}{c} = \frac{\sin(A-B)}{\sin(B-C)}$	$3. \ \Delta$
$(D) \ bc \cos^2 \frac{A}{2}$	$4. \ R r_1 r_2 r_3$
	$5. \ s(s-a)$

If $A, B, C$ are the angles of a triangle,then $\sin^2 A + \sin^2 B + \sin^2 C - 2\cos A \cos B \cos C = $

Similar Questions

The angles $A, B$ and $C$ of a triangle $ABC$ are in $A.P.$ and $a : b = 1 : \sqrt{3}$. If $c = 4 \text{ cm}$,then the area (in $\text{sq. cm}$) of this triangle is:

In a triangle $ABC$,the sides $a, b, c$ are the roots of the equation $x^3-11x^2+38x-40=0$. Then,find the value of $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$.

The number of solutions of the equation $\left( 1 - \frac{1}{2 \sin x} \right) \cos^2 2x = 2 \sin x - 3 + \frac{1}{\sin x}$ in the interval $[0, 4\pi]$ is:

In $\triangle ABC$,if $\cos 3A + \cos 3B + \cos 3C + \cos 3\pi = 0$,then the least value of the sum of two of its angles is

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