In - prove that cos2A2 - cos 2B2 - cos2C2 - 2cosA2cosB2sinC2

Given, cos^2 A2 + cos^2 B2 cos^2 C2 = 2cosA2 . cosB2 . cosC2 Taking L.H.S #160; #160; #160; A + B + C = π = cos^2 A2 + cos^2 B2 cos^2 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard VII

Mathematics

Area of a Triangle

In Δ, prove...

Question

In $Δ$ , prove that ${cos}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} - {cos}^{2} \frac{C}{2} = 2 cos \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}$

Open in App

Solution

Given, ${cos}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} - {cos}^{2} \frac{C}{2} = 2 cos \frac{A}{2} . cos \frac{B}{2} . cos \frac{C}{2}$
Taking L.H.S
$A + B + C = π$
$= {cos}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} - {cos}^{2} \frac{C}{2}$
$= 1 - {sin}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} - {cos}^{2} \frac{C}{2}$
$= 1 - {cos}^{2} \frac{B}{2} - {sin}^{2} \frac{A}{2} - {cos}^{2} \frac{C}{2}$
$= 1 + cos \frac{B + A}{2} . cos \frac{A - B}{2} - {cos}^{2} \frac{C}{2}$
$= 1 + sin \frac{C}{2} . cos \frac{A - B}{2} - 1 + {sin}^{2} \frac{C}{2}$
$= sin \frac{C}{2} [cos \frac{A - B}{2} + sin (\frac{π}{2} - (\frac{A + B}{2})]$
$= sin \frac{C}{2} [cos \frac{A - B}{2} + cos (\frac{A + B}{2}]$
$= sin \frac{C}{2} [2 cos \frac{A}{2} . cos \frac{B}{2}]$
$= 2 cos \frac{A}{2} . cos \frac{B}{2} . sin \frac{C}{2}$

Suggest Corrections

Similar questions

State the whether given statement is true or false

In triangle

A B C,

{cos}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} + {cos}^{2} \frac{C}{2} = 2 cos \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}

Q. If

A + B + C = 180^{0}

, then

(i)

cos 2 A + cos 2 B - cos 2 C = 1 - 4 sin A sin B sin C

(ii)

sin A - sin B + sin C = 4 sin \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}

Prove that: ${sin}^{2} (\frac{A}{2}) + {sin}^{2} (\frac{B}{2}) - {sin}^{2} (\frac{C}{2}) = 1 - 2 cos (\frac{A}{2}) cos (\frac{B}{2}) sin (\frac{C}{2})$ .

Q. In triangle ABC ,prove that

{sin}^{2} \frac{A}{2} + {sin}^{2} \frac{B}{2} - {sin}^{2} \frac{C}{2} = 1 - 2 cos \frac{A}{2} cos \frac{B}{2} s i n \frac{C}{2}

Q. Prove that:

c o s^{2} \frac{A}{2} + c o s^{2} \frac{B}{2} + c o s^{2} \frac{C}{2} = 2 + \frac{r}{2 R}

Join BYJU'S Learning Program

In Δ, prove that cos2A2+cos2B2−cos2C2=2cosA2cosB2sinC2

In $Δ$ , prove that ${cos}^{2} \frac{A}{2} + {cos}^{2} \frac{B}{2} - {cos}^{2} \frac{C}{2} = 2 cos \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}$