Video Transcript
Simplify sin 𝜃 plus cos of 𝜃 all
squared minus two sin 𝜃 cos 𝜃.
In this question, we are asked to
simplify a trigonometric expression. And there are many different
identities and results we can apply. So it is a good idea to look at the
given expression first.
We cannot simplify the first term
using any result. So we will start by evaluating the
exponent to see if this allows us to simplify further. Evaluating the exponent gives us
sin squared 𝜃 plus two sin 𝜃 cos 𝜃 plus the cos squared of 𝜃. We can then subtract the second
term in the given expression. We can note that two sin 𝜃 cos 𝜃
minus two sin 𝜃 cos 𝜃 is equal to zero. This leaves us with sin squared 𝜃
plus cos squared 𝜃.
We can then recall that the
Pythagorean identity tells us that for any angle 𝜃, the sin squared of 𝜃 plus the
cos squared of 𝜃 is equal to one. Hence, we were able to show that
for any value of 𝜃, the sin of 𝜃 plus the cos of 𝜃 all squared minus two sin 𝜃
cos 𝜃 is equal to one.
