1. Using Fundamental Identities
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Recall all of the Reciprocal Identities.
We are asked to match each function with an equivalent expression.
Notice that the expressions labeled with (i), (ii), and (iii) are reciprocals of trigonometric functions. That means we need to recall all of the Reciprocal Identities to match each function with an equivalent expression.
| Reciprocal Identities | ||
|---|---|---|
| sin u = 1/csc u | cos u = 1/sec u | tan u = 1/cot u |
| csc u = 1/sin u | sec u = 1/cos u | cot u = 1/tan u |
Next, let's look for the identities that contain the expressions 1sec u, 1cot u, and 1csc u.
| Reciprocal Identities | ||
|---|---|---|
| sin u = 1/csc u | cos u = 1/sec u | tan u = 1/cot u |
| csc u = 1/sin u | sec u = 1/cos u | cot u = 1/tan u |
According to the first three identities, sin u = 1csc u, cos u = 1sec u, and tan u = 1cot u. With this in mind, we can match each of the functions labeled with (a), (b), and (c) with an equivalent expression.