What is the value of cos squared 67 degrees minus sin squared 23 degrees?
The first thing we should notice here is the relationship between 67 and 23. 67 degrees plus 23 degrees equals 90 degrees, which makes these two angles complementary angles. And here’s what we know about the sine and cosine of complementary angles. Sin of 𝜃 is equal to cos of 90 minus 𝜃.
There’s the same kind of relationship with cosine. Cos of 𝜃 equals sin of 90 minus 𝜃. If we substitute 67 degrees for 𝜃, our statement would say cos of 67 degrees equals sin of 90 degrees minus 67 degrees. Cos of 67 degrees equals sin of 23 degrees.
We can square both sides of the equation. Cos squared of 67 degrees equal sin squared of 23 degrees. If we want to subtract sin squared 23 degrees from cos squared of 67 degrees, the outcome is zero. A value minus the same value equals zero. Cos squared of 67 degrees and sin squared of 23 degrees are the same value. When we subtract them, the result is zero.