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`        Question: What is the minimum value of sin²θ + cos²θ + sec²θ + cosec²θ + tan²θ + cot²θ?Answer:  sin²θ + cos²θ + sec²θ + cosec²θ + tan²θ + cot²θ= (sin²θ + cosec²θ) + (cos²θ + sec²θ) + (tan²θ + cot²θ) Applying AM-GM logic, [i.e. (a + b) ≥ 2 √a*√b]: Minimum value = (2√ sin²θ *√cosec²θ) + (2√ cos²θ *√ sec²θ) + (2√ tan²θ *√ cot²θ)                              = (2√1) + (2√1) + (2√1)                             = 2 + 2 + 2                                = 6Is it correct?`
2 years ago

```							Sorry Avik, that is wrong. Ans:sin2θ + cos2θ = 1sec2θ = 1 + tan2θcosec2θ = 1 + cot2θ Using these identities we can simplify the given equation as   sin²θ + cos²θ + sec²θ + cosec²θ + tan²θ + cot²θ= (sin²θ + cos²θ) + (sec²θ + tan²θ) + (cosec²θ + cot²θ)= 1 + 1 + 2tan²θ + 1 + 2cot²θ= 3 + 2(tan²θ + cot²θ) Now, using AM-GM inequality we can saymin(tan²θ + cot²θ) = 2 So, min(3 + 2(tan²θ + cot²θ)) = 3 + 2*2 = 7
```
2 years ago
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