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Thank you for your prompt answer. But my doubt is not clear. According to the AM-GM inequality: 1)a sin²θ + b cosec²θ ≥ 2 √a*√b 2)a cos²θ + b sec²θ ≥ 2 √a*√b 3)a tan²θ + b cot²θ ≥ 2 √a*√b Applying these inequalities I get answer 6 which is lower than your answer 7. You have applied Pythagorean trigonometric identities first and then AM-GM inequality. So I think 6 is the minimum value. Please explain my mistakes.

Thank you for your prompt answer. But my doubt is not clear. According to the AM-GM inequality:
1)a sin²θ + b cosec²θ   ≥ 2 √a*√b
2)a cos²θ + b sec²θ    ≥ 2 √a*√b
3)a tan²θ + b cot²θ    ≥ 2 √a*√b
Applying these inequalities I get answer 6 which is lower than your answer 7. You have applied Pythagorean trigonometric identities first and then AM-GM inequality. So I think 6 is the minimum value. Please explain my mistakes. 

Grade:12th pass

1 Answers

Utsav Basu
70 Points
6 years ago
Well to clear your doubts, let me just ask you a question:
You claim that
min(sin2θ + cosec2θ) = 2 , and, at the same time,
min(cos²θ + sec²θ) = 2 , right?
Now, is there any value of θ for which your above claims hold simultaneously? The answer is no.
Since, suppose, if min(sin2θ + cosec2θ) = 2, then sinθ = cosecθ = 1
But then, min(cos²θ + sec²θ) = 2 does not hold true.
The same is true for the reverse.
 
I hope I am able to clear your doubts. Please approve my answers if so...

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