Question 1046263: what is the minimum value of sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x) Found 3 solutions by Alan3354, ikleyn, robertb:Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website! sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)
sin^2 + cos^2 = 1
= 1 +sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)
sec^2 = tan^2 + 1
= 1 + tan^2(x) + 1 +cosec^2(x)+tan^2(x)+cot^2(x)
= 2 + 2tan^2(x) + cosec^2(x) + cot^2(x)
csc^2 = cot^2 + 1
= 2 + 2tan^2(x) + cot^2(x)+1 + cot^2(x)
f(x) = 3 + 2tan^2(x) + 2cot^2(x)
f'(x) = 4tan*sec^2 - 4cot*csc^2
tan*sec^2 - cot*csc^2 = 0
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sin/cos^3 - cos/sin^3 = 0
Muliply thru by sin^3*cos^3
sin^4 - cos^4 = 0
(sin - cos)*(sin + cos)*(sin^2 + cos^2) = 0
sin = cos --> x = pi/4 (principal solution)
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sin = -cos --> x = 3pi/4 (principal solution)
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sin^2 = -cos^2 --> no real solution
You can put this solution on YOUR website! what is the minimum value of sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)
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Let me introduce c = cos(x) and s = sin(x) for brevity.
Then
=
= + + + + = ( replace by 1)
= = + + =
= + + = + + = + - = .
Now, = = = has the maximum .
Therefore, has the minimum equal to = 8 - 1 = 7.
Answer. has the minimum of 7.
