The equation whose roots are $sec ^{ 2 }{ \alpha },$ $cosec ^{ 2 }{ \alpha } $ can be

Using the trigonometric identity secant squared alpha equals 1 plus tangent squared alpha and cosecant squared alpha equals 1 plus cotangent squared alpha, the sum of the roots is 2 plus tangent squared alpha plus cotangent squared alpha. Because tangent squared alpha plus cotangent squared alpha is always at least 2, the minimum possible sum of the roots is 4, but we can find a specific equation by checking the options. For the equation x squared minus 9x plus 9 equals 0, the sum of the roots is 9 and the product of the roots is 9. These values are consistent because secant squared alpha times cosecant squared alpha equals 4 divided by the square of sine 2 alpha, which yields 9 when sine 2 alpha is two thirds.

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