Video Transcript
Simplify the sec of 𝜋 over two
minus 𝜃 over cot of 𝜋 minus 𝜃.
In this expression, we have a
reciprocal function divided by a reciprocal function. Additionally, we have a cofunction
identity and a correlated identity. First, the sec of 𝜋 over two minus
𝜃 equals the csc of 𝜃. And second, the cot of 𝜋 minus 𝜃
equals the negative cot of 𝜃. We rewrite sec of 𝜋 over two minus
𝜃 as csc 𝜃. And the cot of 𝜋 minus 𝜃 becomes
the negative cotangent. And then we’ll recall that our
reciprocal functions csc 𝜃 equals one over sin 𝜃, cot 𝜃 equals cos 𝜃 over sin
𝜃.
We’re using a strategy to take all
of our reciprocal functions and write them in terms of sine and cosine, which gives
us one over sin 𝜃 divided by negative cos 𝜃 over sin 𝜃. Dividing by a fraction is
multiplying by its reciprocal. A sine in the denominator and a
sine in the numerator cancel each other out, which equals negative one over cos 𝜃,
which means we’ll need to use one final identity sec 𝜃 equals one over cos 𝜃,
which makes the simplified form negative sec 𝜃.
