Trigonometric Identity Proof

Hi Sagun, let's solve this trigonometric identity together. We need to prove that cosine eighteen degrees minus sine eighteen degrees equals the square root of two times sine twenty seven degrees.

A very useful trick for expressions in the form a cosine theta minus b sine theta is to multiply and divide by the square root of a squared plus b squared.

So, let's rewrite our expression by multiplying and dividing by the square root of two.

Now, we recall that the constant one over the square root of two is equal to both sine forty five degrees and cosine forty five degrees.

Let's substitute these values into our expression. We'll replace the first constant with sine forty five and the second with cosine forty five.

Our expression inside the parentheses now looks like a classic trigonometric identity.