Simplify the sec of 𝜃 multiplied by the tan of 𝜃 multiplied by the tan of 270 degrees plus 𝜃.
When dealing with a problem of this type, it is not always obvious where we should start. As a general rule, it is worth trying to make the argument the same for all parts of the expression. In this question, we will begin by writing the tan of 270 degrees plus 𝜃 simply in terms of 𝜃. One way of doing this is to begin by considering the unit circle as shown. Adding 270 degrees to our angle 𝜃 puts us in the same position on the unit circle as subtracting 90 degrees from 𝜃. This means that the tan of 270 degrees plus 𝜃 is the same as the tan of 𝜃 minus 90 degrees. This looks similar to one of our cofunction identities, which states that the tan of 90 degrees minus 𝜃 is equal to the cot of 𝜃.
If we factor negative one out of the expression in our parentheses, we have the tan of negative 90 degrees minus 𝜃. We recall that the tangent function is odd such that the tan of negative 𝛼 is equal to negative tan 𝛼. The tan of negative 90 degrees minus 𝜃 is therefore equal to negative tan of 90 degrees minus 𝜃. Using the cofunction identity, this is therefore equal to negative cot of 𝜃. Our initial expression, therefore, becomes the sec of 𝜃 multiplied by the tan of 𝜃 multiplied by the negative cot of 𝜃.
Next, we know that one of the reciprocal identities states that cot 𝜃 is equal to one over tan 𝜃. This means that negative cot 𝜃 is equal to negative one over tan 𝜃. We can now cancel the tan 𝜃’s, leaving us with sec 𝜃 multiplied by negative one. This is equal to negative sec 𝜃. As the secant function is the reciprocal of the cosine function, we can also write this as negative one over cos 𝜃. However, in this question, we will leave the expression sec 𝜃 multiplied by tan 𝜃 multiplied by the tan of 270 degrees plus 𝜃 as negative sec 𝜃 in its simplest form.