SOLUTION: what is the minimum value of sin^2(x)+cos^2(x)+sec^2(x)+cosec^2(x)+tan^2(x)+cot^2(x)

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

Answer by ikleyn(52780)   (Show Source): 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.

Plot y =

Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!

= .
Whenever , the relation is always true.
===> The minimum of the original expression is 3 + 2*2 = .
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