Simplify cos 180 degrees minus 𝜃 times cos 90 degrees minus 𝜃 times sec 𝜃 minus 180 degrees.
This expression has three factors and we’re going to simplify each of these three factors before combining them. We start with cos 180 degrees minus 𝜃 which we can evaluate by using the angle difference identity for cos 𝑥 minus 𝑦. Replacing 𝑥 by 180 degrees and 𝑦 by 𝜃, we get that cos 180 degrees minus 𝜃 is equal to cos 180 degrees times cos 𝜃 plus sin of 180 degrees times sin 𝜃. cos of 180 degrees is negative one and sin of 180 degrees is zero. These values are worth remembering. That can be derived using the unit circle. Simplifying the expression using these values, we get that cos 180 degrees minus 𝜃 is equal to minus cos 𝜃.
The next factor, cos 90 degrees minus 𝜃, we could simplify by using the same angle difference identity but instead of going through the hassle, we can just remember the cofunction identity: cos of 90 degrees minus 𝜃 is equal to sin 𝜃.
Our final factor is sec 𝜃 minus 180 degrees. By the definition of the sec function, this is one over cos 𝜃 minus 180 degrees. And we can apply the angle difference identity to the denominator of this fraction. We see cos 180 degrees and sin 180 degrees making an appearance again. We saw before that cos 180 degrees is negative one and sin 180 degrees is zero. Simplifying, we get one over minus cos 𝜃 which is negative sec 𝜃.
Now that we have simplified each factor of our product, we can multiply them together. We get minus cos 𝜃 times sin 𝜃 times minus sec 𝜃. The two minus signs cancel out on multiplication and as sec 𝜃 is one over cos 𝜃, the cos 𝜃 and sec 𝜃 also cancel each other out, and we’re left with sin 𝜃 which we can’t simplify any further. So cos 180 degrees minus 𝜃 times cos 90 degrees minus 𝜃 times sec 𝜃 minus 180 degrees simplifies to just sin 𝜃.
Notice that two of the factors cancelled each other because minus cos 𝜃 times minus sec 𝜃 is one. Could we have seen that cos 180 degrees minus 𝜃 and sec 𝜃 minus 180 degrees cancel out without simplifying them first? That’s something to go away and think about.