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my question is- if in triangle abc, angle c is 120 degrees’ then the value of cos 2 A + cos 2 B – cosA.cosB is { answer is a numerical value}

my question is-
if in triangle abc, angle c is 120 degrees’
then the value of
cos2A + cos2B – cosA.cosB is {answer is a numerical value}
 

Grade:11

1 Answers

Aarushi Ahlawat
41 Points
3 years ago
Since A+B+C = 180 degrees in a triangle. Given C=120 degrees. Hence A+B= 60 Degrees. Now solve the given equation to get (A+B) form and put value of the angle.
cos^2(A)+cos^2(B)-cos(A)cos(B)
=cos^2(A)+1-sin^2(B)-cos(A)cos(B)
=1+cos^2(A)-sin^2(B)-cos(A)cos(B)
=1+cos(A+B)cos(A-B)-cos(A)cos(B)
==1+cos(60^o)\left (cos(A)cos(B)+sin(A)sin(B) \right )-cos(A)cos(B)
=1+\frac{1}{2}\left (cos(A)cos(B)+sin(A)sin(B) \right )-cos(A)cos(B)
=1-\frac{1}{2}\left (cos(A)cos(B)-sin(A)sin(B) \right )
=1-\frac{1}{2}cos(A+B)
=1-\frac{1}{2}\frac{1}{2}
=\frac{3}{4}

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