Question Video: Simplifying Trigonometric Expressions Using a Shift and Reciprocal Identities Mathematics

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.