Question

show that tan² theta + cot ²theta >=2 where theta belongs to real numbers

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Answer 1

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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