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If A and B are complementary acute angles, what is the value of (Tan A + Tan B)/(Csc(360-2A))?

This value is -2 regardless of A and B.

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Borys Shumyatskiy eNotes educator | Certified Educator

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Two complementary acute angles by definition add up to `90^@ , ` i.e. `A + B = 90^@ . ` A place where one can observe two complementary acute angles is a right triangle. Its acute angles plus the right angle `90^@ ` have the sum of measures of `180^@ , ` which is the cause.

In a right triangle, the tangent of an acute angle is the ratio between the opposing leg of the angle and the adjacent leg. Because of this, the tangents of `A ` and `B ` are reverse numbers, `tan B = 1 / tan A .`

Also we know that

`csc ( 360^@ - 2 A ) = 1 / sin ( 360^@ - 2 A ) = 1 / sin ( - 2 A ) = -1 / sin ( 2 A ) . `

The second equality is true because sine is `360^@ `-periodic.

Now combine everything we know:

`( tan A + tan B ) / csc ( 360^@ - 2A ) = ( sin A / cos A + cos A / sin A ) / ( -1 / sin ( 2 A ) ) =` ` ( ( sin^2 A + cos^2 A ) / ( sin A cos A ) ) / ( -1 / ( 2 sin A cos A ) ) = -2 ,`

which is the answer.