Evaluate cos squared 165 degrees minus sin squared 165 degrees without using a calculator.
We’re told we’re not allowed to use a calculator to evaluate this expression. And generally, when we can’t use a calculator to evaluate a trigonometric expression, that means we have to use special angles. Those special angles are zero degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees. And they are called special angles because we know the exact values of their sine and cosine. One way to remember these sines as special angles is you count up from zero to four. We then square root each of these values and divide them by two.
So, we see for example that sin of zero degrees is the square root of zero divided by two, which is just zero. Sin 30 degrees is the square root of one divided by two, which is a half. Sin of 45 degrees and 60 degrees, we can’t simplify any further. But as the square root of four is two, sin of 90 degrees becomes two over two, which is one. And we get the same values in the opposite order for cos 𝜃.
Now that’s all well and good. But our expression involves cos of 165 degrees and sin of 165 degrees. And 165 degrees isn’t one of our special angles. We can’t just substitute in the values of cos 165 degrees and sin 165 degrees. If we want to use special angles, we’re first going to have to transform our expression. The trick to doing this is to notice that our expression is of the form cos squared 𝑥 minus sin squared 𝑥, where 𝑥 is 165 degrees.
And, having written this down, we might recognize this from the identity for cos two 𝑥. For any value of 𝑥, cos two 𝑥 is cos squared 𝑥 minus sin squared 𝑥. And so, swapping the sides, cos squared 𝑥 minus sin squared 𝑥 is cos two 𝑥. Applying this identity, then, we see that our expression becomes cos of two times 165 degrees, which is cos of 330 degrees. Now we have a much simpler expression. But 330 degrees still isn’t one of our special angles. We need to use some more identities.
First, we can use the fact that cos is a periodic function with period 360 degrees. And so cos of 360 degrees plus 𝑥 is just the same as cos 𝑥. By repeatedly applying this identity, we can see that if we add any multiple of 360 degrees to the argument of cosine, we don’t change the value of the cosine. So, if we write 330 degrees as 360 degrees minus 30 degrees, we can subtract 360 degrees from the argument of cosine to leave just cos of negative 30 degrees. Now we’re getting closer. 30 degrees is a special angle. And so we know the value of cos of 30 degrees.
Unfortunately, we have this minus sign here. But, again, an identity comes to the rescue. We can use the fact that cos is an even function and hence cos of negative 𝑥 is just cos of 𝑥. And cos of negative 30 degrees is cos of 30 degrees. Now we can use our table to just read off the value of cos 30 degrees. It’s root three over two. And so, that’s our final answer. Cos squared of 165 degrees minus sin squared 165 degrees is root three over two. And we found this value without using a calculator.
To recap, then, our first step and the main trick in solving this question was to use the double-angle identity for cosine. After some simplification, we used the fact that the cosine function, cos, is a periodic function and an even function, leaving us with just cos 30 degrees. And 30 degrees is a special angle. And so we knew the value of cos 30 degrees. It’s our final answer: root three over two.