Question

sin+cosec=2so what happen sinsin2+cosec2???

Answer 1

Answer:

Step-by-step explanation:

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Use the AM ≥ GM, inequality for sin2x and cosec2x:

sin2x+cosec2x2≥(sin2x.cosec2x)−−−−−−−−−−−−−√

⟹sin2x+cosec2x2≥1 , since (sin2x.cosec2x)=1

So what does it tell about the maximum value of sin2x+cosec2x ?

Yes absolutely, this expression doesn't have a maximum value. But yes the minimum value is 2.

Another way to see this is by considering the function

f(x)=sin2x+cosec2x=(sinx−cosecx)2+2.sinx.cosecx=(sinx−cosecx)2+2=(sinx−1sinx)2+2=2+(−cos2xsinx)=2+cot2xcosec2x

Since, both cotx and cosecx takes infinitely large values therefore the function f(x) also takes infinitely large values, implying f has no maximum value