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how can I calculate average of cos^{2}x ?

I want to take average over a sphere

I tried to do like this: <cos^{2}X>= 1/2π ∫cos^{2}xdx

and I get 1/2

but in my books, wrote that average of cos^{2}x , taken over a sphere, is 1/3

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# Average of COSIN

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