`sec(v - u)` Find the exact value of the trigonometric expression given that sin(u) = 5/13 and cos(v) = -3/5 (both u and v are in quadrant II.)

There are a lot of ways to solve this problem. For instance, one can use the secant sum/difference identity. This identity, however, has a lot of terms. A more elegant solution would be to use the cosine sum/difference identity, and relating cosine to secant.

Note that `sec(x) = 1/(cos(x))` .Such that we only need to get the reciprocal of the value of cosine at our given angle to get the corresponding secant. Both u and...

There are a lot of ways to solve this problem. For instance, one can use the secant sum/difference identity. This identity, however, has a lot of terms. A more elegant solution would be to use the cosine sum/difference identity, and relating cosine to secant.

Note that `sec(x) = 1/(cos(x))` .Such that we only need to get the reciprocal of the value of cosine at our given angle to get the corresponding secant. Both u and v are in the second quadrant implying that the x-components are both negative and the y-components positive. 

The cosine sum/difference identity is: `cos(v-u) = cos(v)cos(u) + sin(v)sin(u)` .

We already know some of the terms here, namely: `sin(u) = 5/13` and `cos(v) = -3/5` .To get the remaining terms, we simply use the Pythegorean identity: `sin^2(x) + cos^2(x) = 1` .

Hence:

`cos^2(u) = 1 - 25/169 = 144/169` , and `cos(u) = -12/13` . Note that we could have chosen the positive root, but since u is in the second quadrant, we get the negative one.

Similarly:

`sin^2(v) = 1 - 9/25 = 16/25` , and `sin(v) = 4/5` . We are getting the positive root since v is in the second quadrant.

Now, we can solve for the cosine:

 `cos(v-u) = (-3/5)(-12/13) + (4/5)(5/13) = 36/65 + 20/65 = 56/65.`

Which means:

`sec(v-u) = 65/56.`