(D) We use the identity $\cos^2 \theta = \frac{1+\cos(2\theta)}{2}$.Thus,$\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2} = \frac{1+\cos A

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(D) We use the identity $\cos^2 \theta = \frac{1+\cos(2\theta)}{2}$.Thus,$\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2} = \frac{1+\cos A}{2} + \frac{1+\cos B}{2} + \frac{1+\cos C}{2}$.$= \frac{1}{2} \{3 + (\cos A + \cos B + \cos C)\}$.Using the identity $\cos A + \cos B + \cos C = 1 + \frac{r}{R}$,where $r$ is the inradius and $R$ is the circumradius of the triangle:$= \frac{1}{2} \{3 + 1 + \frac{r}{R}\}$.$= \frac{1}{2} \{4 + \frac{r}{R}\}$.$= 2 + \frac{r}{2R}$.

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

Easy

In any triangle $ABC$,find the value of $\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2}$.

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A
$1+\frac{r}{2R}$
B
$2-\frac{r}{2R}$
C
$2+\frac{r}{4R}$
D
$2+\frac{r}{2R}$

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Trigonometrical Equations

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

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In any triangle $ABC$,find the value of $\cos ^2 \frac{A}{2}+\cos ^2 \frac{B}{2}+\cos ^2 \frac{C}{2}$.

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