Question Video: Simplifying Trigonometric Expressions Using Shift, Cofunction and Pythagorean Identities Mathematics • First Year of Secondary School

Video Transcript

Simplify one plus cot squared three 𝜋 over two minus 𝜃 over one plus tan squared 𝜋 over two minus 𝜃.

In order to answer this question, we will need to use a variety of trigonometric identities. There are many ways to start here. However, we will begin by trying to rewrite the expression simply in terms of 𝜃. By firstly sketching the unit circle, we recall that 𝜋 radians is equal to 180 degrees. This means that 𝜋 over two radians is equal to 90 degrees. The denominator of our expression can therefore be rewritten as one plus tan squared of 90 degrees minus 𝜃. One of our cofunction identities states that tan of 90 degrees minus 𝜃 is equal to cot 𝜃. This means that tan squared of 90 degrees minus 𝜃 is equal to cot squared 𝜃. And the denominator of our expression is therefore equal to one plus cot squared 𝜃.

Let’s now consider the angle three 𝜋 over two minus 𝜃. Once again, we can see from the unit circle that three 𝜋 over two radians is equal to 270 degrees. This means that the numerator of our expression is equal to one plus cot of 270 degrees minus 𝜃. If 𝜃 lies in the first quadrant, as shown in our right triangle, then three 𝜋 over two minus 𝜃, or 270 degrees minus 𝜃, lies in the third quadrant. It is clear from the diagram that cos of three 𝜋 over two minus 𝜃 is equal to negative sin 𝜃 and sin of three 𝜋 over two minus 𝜃 is equal to negative cos 𝜃. Since sin 𝜃 over cos 𝜃 is tan 𝜃 and cos 𝜃 over sin 𝜃 is cot 𝜃, then cot of 270 degrees minus 𝜃 is equal to tan 𝜃. Squaring both sides of this identity, we can rewrite the numerator of our expression as one plus tan squared 𝜃.

Our next step is to recall two of the Pythagorean identities. Firstly, tan squared 𝜃 plus one is equal to sec squared 𝜃. And secondly, one plus cot squared 𝜃 is equal to csc squared 𝜃. Our expression simplifies to sec squared 𝜃 over csc squared 𝜃. And this can be rewritten as sec squared 𝜃 multiplied by one over csc squared 𝜃. Recalling the reciprocal identities sec 𝜃 is equal to one over cos 𝜃 and csc 𝜃 is equal to one over sin 𝜃, we have one over cos squared 𝜃 multiplied by sin squared 𝜃, which can be rewritten as sin squared 𝜃 over cos squared 𝜃 and, in turn, is equal to tan squared 𝜃. The expression one plus cot squared three 𝜋 over two minus 𝜃 over one plus tan squared 𝜋 over two minus 𝜃 written in its simplest form is tan squared 𝜃.