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Find sec 𝜃 − tan 𝜃 given sec 𝜃 + tan 𝜃 = −14/27.

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.

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