if A+B=60 then cos^2A+cos^B-cosAcosB= ?
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3 ANSWERS

Bitan Sarkar, Btech Electronics and Communication Engineering, Indian Institute of Technology, Bhubaneswar (2021)
Answered Jun 1, 2017
If I need to find the value very quickly, then I would prefer putting [math]A = 60° and B = 0°[/math]
[math]cos^2A + cos^2B - cosAcosB = 1 + 1/4 -1/2 = 3/4[/math]
Now take [math]A = 30° and B = 30°[/math]
[math]cos^2A + cos^2B - cosAcosB = 3/4 + 3/4 - 3/4[/math]
Thus I can roughly say, yes [math]cos^2A + cos^2B - cosAcosB = 3/4[/math], for [math]A + B = 60°[/math]
Now, if you are looking for a proper method,
then also its easy,
[math]A + B = 60°[/math]
[math]=> cos(A + B) = 1/2[/math]
[math]=> cosA.cosB - sinA.sinB = 1/2[/math]
[math]=> 2cosA.cosB - 1 = 2sinA.sinB[/math]
squaring both sides,
[math]=> 4cos^2A.cos^2B -4cosA.cosB + 1 = 4sin^2A.sin^2B[/math]
[math]=> 4cos^2A.cos^2B -4cosA.cosB + 1 = 4(1 - cos^2A)(1 - cos^2B)[/math]
[math]=> 4cos^2A.cos^2B -4cosA.cosB + 1 = 4 - 4cos^2A - 4cos^2B + 4cos^2A.cos^2B[/math]
[math]=> -4cosA.cosB = 3 - 4cos^2A - 4cos^2B[/math]
Thus
[math]4cos^2A + 4cos^2B -4cosA.cosB = 3[/math]or [math]cos^2A + cos^2B -cosA.cosB = 3/4[/math]
Answer:
cos²A + cos²B - cosA cosB = 3/4
Step-by-step explanation:
Given A+B=60
A = 60 - B
Apply cos on both sides
cos(A) = cos(60 - B)
cos(A - B) = cosAcosB + sinAsinB
cosA = cos60cosB + sin60sinB
cos60 = 1/2, sin60 =√3/2
cosA = 1/2 cosB + √3/2 sinB
2cosA = cosB + √3 sinB
2cosA - cosB = √3 sinB
Squaring on both the sides.
(2cosA - cosB)² = (√3 sinB)²
4cos²A + cos²B - 4cosA cosB = 3sin²B
here we can write sin²B = 1 - cos²B
4cos²A + cos²B - 4cosA cosB = 3(1 - cos²B)
4cos²A + cos²B - 4cosA cosB = 3 - 3cos²B
4cos²A + cos²B + 3cos²B - 4cosA cosB = 3
4cos²A + 4cos²B - 4cosA cosB = 3
cos²A + cos²B - cosA cosB = 3/4
therefore the value of cos²A + cos²B - cosA cosB = 3/4.
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