show that tan² theta + cot ²theta >=2 where theta belongs to real numbers
Answers
To prove :
tan^2 theta + cot^2 theta >= 2, where theta belongs to real numbers.
Proof :
tan theta = sin theta / cos theta and cot theta = cos theta / sin theta
tan ^2 theta :
> [ sin theta / cos theta ]^2
> [ sin^2 theta / cos ^2 theta ]
cot^2 theta :
> [ cos theta/sin theta ]^2
> [ cos^2 theta / sin^2 theta ]
tan^2 theta + cot^2 theta
> (sin^2 theta/cos^2 theta) + (cos^2 theta/sin^2 theta)
> ( sin^4 theta + cos^4 theta )/( sin^2 theta cos^2 theta )
> [ ( sin^2 theta + cos^2 theta )^2 - 2 sin^2 theta cos^2 theta ]/[ sin^2 theta cos^2 theta ]
> [ 1 - 2 sin^2 theta cos^2 theta ]/[ sin^2 theta cos^2 theta ]
> 1/sin^2 theta cos^2 theta - 2
1/sin^2 theta cos^2 theta >= 4 [ proof by am gm ]
Hence, tan^2 theta + cot^2 theta >= 2
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