Home
Class 12
MATHS
If A+B+C=0 , Prove :cos^2 A + cos^2 B +c...

If` A+B+C=0` , Prove :`cos^2 A + cos^2 B +cos^2 C=1+2cosA cosB cosC`.

Text Solution

Verified by Experts

(a) `LHS=cos^(2)A+cos^(2)A+cos^(2)B+cos^(2)C` ltbr. `=(3+(cos 2A+cos2B+cos2C))/(2)`
now given `A+B=C`
or `A+B+(pi-C)=pi`
or `A+B+D=pi`, where `D=pi-C`
`rArr LHS=(3+(cos2A+cos2B+cos2C))/(2)`
`=(3+(Cos2A+Cos2B+cos2D))/(2)`
`=1-2cos A cos B cos D`
`=1-2cos A cos B cos (pi-C)`
`=1+2cos A cos B cos C=RHS`.
(b) Given `alpha+beta=60^(@)`
`cos^(2)alpha+cos^(2)beta-cos alpha cos beta`
`=1-sin^(2)alpha+cos^(2)beta-cos alpha cos beta`
`=1+cos(alpha+beta)cos(alpha-beta)-cos alpha cos beta`
`=1+cos60^(@)cos(alpha-beta)-cos alpha cos beta`
`=1+(cos alpha cos beta+sin alpha sin beta)/(2)-cos alpha cos beta`
`=1-(cos(alpha+beta))/(2)`
`=1-(1)/(4)=(3)/(4)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If A+B+C=180^@ , then prove that cos^2 A + cos^2 B +cos^2 C=1-2cosA cosB cosC .

If A+B+C=pi , prove that : cos2A+cos2B+cos2C=-1-4cosA cosB cosC

If A+B+C = pi , prove that : cos2A +cos2B +cos2C =-1-4cosAcosBcosC .

If A+B+C=2S , prove that : cos^2 S + cos^2 (S-A) + cos^2 (S-B) + cos^2 (S-C) = 2+2cosA cosB cosC .

If A+B+C=180^0 , prove that : sin^2 A + sin^2 B + sin^2 C =2 (1+cosA cosB cosC)

If A+B+C=2pi , prove that : cos^2B+cos^2C-sin^2A-2cosA cosB cosC=0 .

In a triangle ABC, prove that: cos^4A+cos^4B+cos^4C= 3/2 + 2 cosA cosB cosC+ 1/2 cos 2A cos2B cos2C

If A+B+C=pi/2 , prove that: sin2A + sin2B+sin2C = 4cosA cosB cosC

If A+B+C=pi , prove that : cosA + cosB-cosC=4cos(A/2) cos(B/2) sin(C/2) -1

If A+B+C=180^0 , prove that : cos^2 (A/2) + cos^2 (B/2) - cos^2 (C/2) = 2cos (A/2) cos (B/2) sin (C/2)