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How is cosec^2(pi+pi/6) = (-cosec pi/6)^2 ???

We are asked to show that `csc^2(pi+pi/6)=(-csc (pi/6))^2 ` :


First, the cosecant function has a period of 2pi; that is adding or subtracting multiples of 2pi to the argument leaves the result unchanged.


If instead we look at the square of the cosecant function, we notice that the period is now pi units. If you look at the graph of the cosecant function there are parts above the x-axis and below the x-axis. The parts below are a glide reflection of the parts above. Squaring results in all parts of the graph above the x-axis with translational symmetry of pi units. (See graphs.)


Thus `csc^2(pi+pi/6)=csc^2(pi/6) `


The cosecant function returns a real value for all inputs in its domain, and for real numbers `(-a)^2=a^2 ` . So `(-csc (pi/6))^2=(csc(pi/6))^2 `


But the notations `(csc(pi/6))^2"and"csc^2(pi/6) ` are interchangeable by convention.


Thus ` csc^2(pi+pi/6)=csc^2(pi/6)=(csc(pi/6))^2=(-csc(pi/6))^2 `


The graph of the cosecant function in blue, and the square of the function in red:



Note that shifting pi (about 3.14) units either left or right results in the same y-value in the squared (red) graph.

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