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Find the value of sec 105°/csc 15°.

Find the value of sec 105 degrees divided by csc of 15 degrees.

In order to answer this question, we will use a variety of trigonometric identities. Two of the reciprocal trigonometric identities state that sec 𝜃 is equal to one over cos 𝜃 and csc 𝜃 is equal to one over sin 𝜃. This means that the numerator of our expression is the same as one over the cos of 105 degrees. And the denominator is equal to one over sin of 15 degrees. The expression sec of 105 degrees over csc of 15 degrees can therefore be rewritten as sin of 15 degrees over the cos of 105 degrees. 105 degrees is equal to 90 degrees plus 15 degrees, which can also be written as 90 degrees minus negative 15 degrees.

From our knowledge of complementary angles, one of the cofunction identities states that the cos of 90 degrees minus 𝜃 is equal to sin 𝜃. This means that the cos of 90 degrees minus negative 15 degrees is equal to the sin of negative 15 degrees. Our expression simplifies to sin of 15 degrees over sin of negative 15 degrees. As sin 𝜃 is an odd function, the sin of negative 𝜃 is equal to negative sin 𝜃. This means that the denominator can be rewritten as negative sin of 15 degrees. Dividing the numerator and denominator of our fraction by the sin of 15 degrees gives us one over negative one. And this is equal to negative one. sec of 105 degrees divided by csc of 15 degrees is equal to negative one.

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