(C) Given that $a, b, c$ are in $A.P.$,we have $2b = a + c$. The expression is $E = a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(C) Given that $a, b, c$ are in $A.P.$,we have $2b = a + c$. The expression is $E = a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right)$. Using the identity $2 \cos^2\theta = 1 + \cos(2\theta)$,we get: $E = \frac{a}{2}(1 + \cos C) + \frac{c}{2}(1 + \cos A)$ $E = \frac{1}{2}(a + a \cos C + c + c \cos A)$ By the projection formula,$a \cos C + c \cos A = b$. $E = \frac{1}{2}(a + c + b)$ Since $a + c = 2b$,we have: $E = \frac{1}{2}(2b + b) = \frac{3b}{2}$.

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

Easy

In $\triangle ABC$,with usual notations,if $a, b, c$ are in $A.P.$,then $a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right) = $

MHT CET 2018 All MHT CET

A
$\frac{3a}{2}$
B
$\frac{3c}{2}$
C
$\frac{3b}{2}$
D
$\frac{3abc}{2}$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

Trigonometrical Equations

Subtopic

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

Previous Year

MHT CET 2018 Paper All MHT CET years →

In a triangle $ABC$,with usual notations,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

View Solution

The circumradius $R$ of an isosceles triangle $ABC$ is four times its inradius $r$. If $A = B$,then which of the following is true?

Difficult

View Solution

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

View Solution

In $\triangle ABC$,with usual notations,$m \angle C = \frac{\pi}{2}$. If $\tan \left(\frac{A}{2}\right)$ and $\tan \left(\frac{B}{2}\right)$ are the roots of the equation $a_1 x^2 + b_1 x + c_1 = 0$ $(a_1 \neq 0)$,then:

MediumMHT CET 2023

View Solution

In a triangle,the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$,where $c$ is the third side of the triangle,then the ratio of the in-radius to the circum-radius of the triangle is

IIT 2014

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

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

In $\triangle ABC$,with usual notations,if $a, b, c$ are in $A.P.$,then $a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right) = $

Similar Questions

In a triangle $ABC$,with usual notations,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 circumradius $R$ of an isosceles triangle $ABC$ is four times its inradius $r$. If $A = B$,then which of the following is true?

In $\triangle ABC$,with usual notations,$m \angle C = \frac{\pi}{2}$. If $\tan \left(\frac{A}{2}\right)$ and $\tan \left(\frac{B}{2}\right)$ are the roots of the equation $a_1 x^2 + b_1 x + c_1 = 0$ $(a_1 \neq 0)$,then:

In a triangle,the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$,where $c$ is the third side of the triangle,then the ratio of the in-radius to the circum-radius of the triangle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module