The long notation involving degrees, minutes and seconds does make 
the problem confusing.

So let's first look at and do a similar problem that just uses something
simpler, say, 20°.  For instance let's do this problem:

The equation would come from this.

 

Then 

x = the desired angle
90°-x = the complement of x

x = (90°-x)-20°       <-- step 1
x = 90°-x-20°         <-- step 2
x = 70°-x             <-- step 3 
2x = 70°              <-- step 4
 x = 35°              <-- step 5

So to do our problem we use this as a model to go by.

let's put 18°2'10" in place of 20° in step 1, and then do step 2:

x = (90°-x)-18°2'10"  <-- step 1
x = 90°-x-18°2'10"    <-- step 2

To get step 3 we need to subtract 18°2'10" from 90°:

We start by writing 90° as 89°59'60".
Then we subtract:

 89°59'60"
-18° 2'10"
 71°57'50"

Now we have something to use in place of the 70° in step 3 of the
other problem.

x = 71°57'50"-x       <-- step 3 

Then step 4 is easy:

2x = 71°57'50"        <-- step 4

Now to get the last step we have to divide both sides by 2.  That presents a
difficulty, because 71 is not divisible by 2, so we have to convert it so that
the number of degrees is divisible by 2.  We take 1° away from the 71° to
make it something that is divisible by 2, namely 70°, then we change the 1° to
60' and add that to the 57' getting 117' and we have:

2x = 70°117'50"       <-- step 4

But we still have a problem! 117' is not divisible by 2. Now we must convert the
minutes so that they are divisible by 2.  So we take 1' away from the 117' to
make the number of minutes divisible by 2, namely 116', then we change the 1'
that we took away to 60" and add that to the 50" getting 110" and we have:

2x = 70°116'110"      <-- step 4

Finally step 5 is easy:

 x = 35°58'55"        <-- step 5

Edwin