Question

Minimum value of sintheta+cosec theta whole square +cos theta +sec theta whole square

Answer 1

it is given that (sinθ + cosecθ)² + (cosθ + secθ)²

we have to find the minimum value of above expression.

(sinθ + cosecθ)² + (cosθ + secθ)²

= sin²θ + cosec²θ + 2sinθ. cosecθ + cos²θ + sec²θ + 2cosθ. secθ

we know, cosecθ. sinθ = 1 and cosθ. secθ = 1

= (sin²θ + cosec²θ) + 2 + (cos²θ + sec²θ) + 2

= (sin²θ + cosec²θ) + (cos²θ + sec²θ) + 4

we know, sin²θ , cosec²θ are positive

so AM ≥ GM

(sin²θ + cosec²θ)/2 ≥ √(sinθ. cosecθ)

⇒sin²θ + cosec²θ ≥ 2

so minimum value of sin²θ + cosec²θ = 2

similarly you can get minimum value of cos²θ + sec²θ = 2

now minimum value of (sin²θ + cosec²θ) + (cos²θ + sec²θ) + 4

= 2 + 2 + 4

= 8

hence minimum value of (sinθ + cosecθ)² + (cosθ + secθ)² = 8

Answer 2

Answer:

ans is 8

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