Simplify one minus the cos squared of 𝜃 divided by the sin squared of 𝜃 minus one in its simplest form.
Looking at this fraction, something that can stand out to us is that, in the numerator, we have a cosine squared term and, in the denominator, we have a sine squared term. This can remind us of the Pythagorean identity that the sine squared of any angle plus the cosine squared to that same angle equals one. This implies that the sin squared of 𝑥 equals one minus the cos squared of 𝑥. And the cos squared of 𝑥 equals one minus the sin squared of 𝑥. We note this because these relationships point to two substitutions we can make into our original fraction.
We can replace cos squared 𝜃 with one minus sin squared 𝜃 and sin squared 𝜃 with one minus cos squared 𝜃. Working out the correct signs in numerator and denominator, we have one minus one in our numerator and one minus one in the denominator. We have then negative sin squared 𝜃 over cos squared 𝜃. And now we can recall that the sine of a given angle divided by the cosine of that same angle equals the tangent of that angle. And that means that the sin squared of 𝑥 over the cos squared of 𝑥 equals the tan squared of 𝑥.
Therefore, we can write our fraction of sine squared over cosine squared simply as tangent squared. And we now have the simplest form of our original fraction. One minus the cos squared of 𝜃 divided by the sin squared of 𝜃 minus one is equal to negative tan squared of 𝜃.