Simplify sin of 𝜋 over two plus 𝜃 multiplied by sec of negative 𝜃.
There are many ways of simplifying this expression. In this question, we will look at each factor separately. We will then simplify each of them using the cofunction identities together with the odd and even identities. Let’s begin by considering the sin of 𝜋 over two plus 𝜃. We know that 𝜋 over two radians is equal to 90 degrees. One of our cofunction identities states that the sin of 90 degrees minus 𝜃 is equal to cos 𝜃. Our expression is the sum of two angles and not the difference. However, we can rewrite this as the sin of 𝜋 by two minus negative 𝜃. Using the cofunction identity, this is therefore equal to the cos of negative 𝜃.
We know that cos 𝜃 is an even function such that cos of negative 𝜃 is equal to cos of 𝜃. This means that the sin of 𝜋 over two plus 𝜃 can be simplified to cos 𝜃. We will now focus on the second part of our expression. The secant function is the reciprocal of the cosine function such that sec 𝜃 is equal to one over cos 𝜃. The sec of negative 𝜃 is therefore equal to one over the cos of negative 𝜃. And using the fact that cosine is an even function, once again, this is equal to one over cos 𝜃.
We now have simplified expressions for both parts of the initial expression. The sin of 𝜋 over two plus 𝜃 multiplied by the sec of negative 𝜃 is equal to cos 𝜃 multiplied by one over cos 𝜃, which is equal to one. The expression sin of 𝜋 over two plus 𝜃 multiplied by the sec of negative 𝜃 is equal to one.