4 (bc ^2 A 2 +ca ^2 B 2 +ab ^2 C 2 )=(a + b + c)^2 - Vidyadip

Question

$4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)=(\text{a + b + c})^2$

✓

Answer

$4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)=(\text{a + b + c})^2$$\text{LHS}=4\Big(\text{bc}\cos^2\frac{\text{A}}{2}+\text{ca}\cos^2\frac{\text{B}}{2}+\text{ab}\cos^2\frac{\text{C}}{2}\Big)$
$=2\Big(\text{bc.}2\cos^2\frac{\text{A}}{2}+\text{ca.}2\cos^2\frac{\text{B}}{2}+\text{ab}.2\cos^2\frac{\text{C}}{2}\Big)$
$=2(\text{bc.}(1-\cos\text{A})+\text{ca.}(1-\cos\text{B})+\text{ab.}(1-\cos\text{C}))$
$=2\text{bc}-2\text{bc}\cos\text{A}+2\text{ca}-2\text{ca}\cos\text{B}+2\text{ab}-2\text{ab}\cos\text{C}$
$=2\text{bc}-2\text{bc}\frac{\text{b}^2+\text{c}^2-\text{a}^2}{2\text{bc}}+2\text{ca}-2\text{ca}\frac{\text{a}^2+\text{c}^2-\text{b}^2}{2\text{ca}}+2\text{ab}$
$-2\text{ab}\frac{\text{b}^2+\text{a}^2-\text{c}^2}{2\text{ab}}\text{[cos rule]}$
$=2\text{bc}-\text{b}^2-\text{c}^2+\text{a}^2+2\text{ca}-\text{a}^2-\text{c}^2+\text{b}^2+2\text{ab}-\text{b}^2-\text{a}^2+\text{c}^2$
$=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{ab}+2\text{ca}$
$=(\text{a + b + c})^2=\text{RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Sine And Cosine Formulae And Their Applications→Sine And Cosine Formulae And Their Applications — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

In the triangle ABC with vertices A(2, 3), B (4, -1) and C (1, 2) find the equation and length of altitude from the vertex A.

In how many ways can one select a circket team of eleven from17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?

Find the center, eccentricity, foci and directrices of the hyperbola
$\text{x}^{2}-\text{y}^{2}+4\text{x}=0$

Prove the following by the principle of mathematical induction:
$1 + 2 + 3 + ... + \text{n}=\frac{\text{n}(\text{n}+1)}{2}$ i.e, the sum of the first n natural numbers is $\frac{\text{n}(\text{n}+1)}{2}.$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\sqrt{10}}}\frac{\sqrt{7+2\text{x}}-\big(\sqrt{5}+\sqrt{2}\big)}{\text{x}^2-10}$

Show by the Principle of Mathematical induction that the sum $S_n$ of the n terms of the series $1^2 + 2 \times 2^2 + 3^2 + 2 \times 4^2 + 5^2 + 2 \times 6^2 + 7^2 + ...$ is given by
$\text{S}_\text{n}=\begin{cases}\frac{\text{n}(\text{n}+1)^2}{2},\text{if n is even}\\\frac{\text{n}^2(\text{n}+1)}{2},\text{if n is odd}\end{cases}$

Find the length of the perpendicular from the point (4, -7) to the line joining the origin and the point of intersection of the lines 2x - 3y + 14 = 0 and 5x + 4y - 7 = 0.

Prove that:
$\cos3\text{A}+\cos5\text{A}+\cos7\text{A}+\cos15\text{A}=4\cos4\text{A}\cos5\text{A}\cos6\text{A}$

If $\cos\alpha+\cos\beta=0=\sin\alpha+\sin\beta,$ then prove that $\cos2\alpha+\cos2\beta=-2\cos(\alpha+\beta).$
$\big[$Hint: $(\cos\alpha+\cos\beta)^2-(\sin\alpha+\sin\beta)^2=0\big]$

Find the standard deviation for the following data:

$x$	$3$	$8$	$13$	$18$	$23$
$f$	$7$	$10$	$15$	$10$	$6$