c. Show that

Let’s dive into some trigonometric fun! To show that , we can use the angle addition and double angle formulas.

Start by using the sine addition formula:

Now recall the double angle identities:

Substituting these into our formula gives us:

This simplifies to:

Since , we can rewrite this as:

Now substitute , leading us to:

Voilà! The identity is proven.

Now, as a little extra sprinkle of math magic, did you know that trigonometric identities like this one pop up in various fields, from physics to engineering? For instance, they help in understanding wave behavior, especially in AC circuits, where sine waves represent voltage and current variations over time. How cool is it to use an ancient math idea in modern tech?

For further exploration, you can dive deeper into the world of trigonometric identities by checking out resources like “Trigonometry” by I.M. Gelfand and Mark Saul, or even online platforms like Khan Academy where they break down these topics with visual aids and interactive exercises. You’ll be a math wizard in no time!