### Video Transcript

Find the sec of 𝜃 minus the tan of
𝜃 given the sec of 𝜃 plus the tan of 𝜃 is equal to negative 14 over 27.

We recall that the difference of
two squares states that 𝑥 squared minus 𝑦 squared is equal to 𝑥 plus 𝑦
multiplied by 𝑥 minus 𝑦. This means that sec 𝜃 plus tan 𝜃
multiplied by sec 𝜃 minus tan 𝜃 is equal to sec squared 𝜃 minus tan squared
𝜃. We are told in the question the
value of sec 𝜃 plus tan 𝜃. It is equal to negative 14 over
27. And we need to calculate the value
of sec 𝜃 minus tan 𝜃.

One of the Pythagorean identities
states that tan squared 𝜃 plus one is equal to sec squared 𝜃. If we subtract tan squared 𝜃 from
both sides of this equation, we have sec squared 𝜃 minus tan squared 𝜃 is equal to
one. This is equivalent to the
right-hand side of our equation. Negative 14 over 27 multiplied by
sec 𝜃 minus tan 𝜃 is equal to one. We can then divide both sides of
this equation by negative 14 over 27. We know that dividing by a fraction
is the same as multiplying by the reciprocal of this fraction. Therefore, the right-hand side is
equal to one multiplied by negative 27 over 14.

If the sec of 𝜃 plus the tan of 𝜃
is equal to negative 14 over 27, then the sec of 𝜃 minus the tan of 𝜃 is equal to
negative 27 over 14. These values are the reciprocal of
one another.