wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

In a ΔABC, 11+tan2A2+11+tan2B2+11+tan2C2=k[1+sinA2sinB2sinC2], then the value of k is

Open in App
Solution

LHS=11+tan2A2+11+tan2B2+11+tan2C2
=cos2A2+cos2B2+cos2C2
=1+cosA2+1+cosB2+1+cosC2
=32+12(cosA+cosB+cosC)
=32+12[12sin2A2+2cos(B+C2)cos(BC2)]
=32+12[12sin2A22sinA2cos(BC2)]
=2sinA2[sinA2cos(BC2)]
=2sinA2[cos(B+C2)cos(BC2)]
=2+2sinA2sinB2sinC2
=2[1+sinA2sinB2sinC2]
​​​​​​​
RHS=k[1+sinA2sinB2sinC2]
k=2


Alternate solution:
If we assume A=B=C=60, we get
11+tan2A2+11+tan2B2+11+tan2C2
=cos2A2+cos2B2+cos2C2=cos230+cos230+cos230=3×34=94

k[1+sinA2sinB2sinC2]=k[1+(sin30)3]=k[1+18]=9k8
Now,
9k8=94k=2

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Conditional Identities
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon