Write the expression as a single trigonometric functions:

cos 6x cosx - sin6x sin x

if you could do it step by step?

also... How do I prove that sin(90deg + theta) = cos(theta)??

also step by step so i may learn?

the important identity here is

cos(A+B) = cosAcosB - sinAsinB

So, it looks like you have

cos(6x+x) = cos 7x

You also know that

sin(A+B) = sinAcosB + cosAsinB
so,
sin(90+x) = sin90cosx + cos90sinx = 1*cosx = 0*sinx = cosx

Do lots of these, since they will come in handy later on.

Sure! Let's break down both problems step by step.

1. Writing the expression as a single trigonometric function:
To simplify the given expression, we can use the trigonometric identity called the product-to-sum formula, which states that cos A cos B - sin A sin B is equal to cos(A + B).
Let's apply this formula to the given expression:

cos 6x cos x - sin 6x sin x

Using the product-to-sum formula, we can rewrite it as:

= cos(6x + x)

= cos(7x)

So, the simplified expression is cos(7x).

2. Proving that sin(90° + θ) = cos θ:
To prove this trigonometric identity, we can use the angle addition formula for sine, which states that sin(A + B) = sin A cos B + cos A sin B.
Let's use this formula to prove the given identity step by step:

sin(90° + θ)

Using the angle addition formula, we can rewrite it as:

= sin 90° cos θ + cos 90° sin θ

Now, let's evaluate the sine and cosine of 90°:

sin 90° = 1
cos 90° = 0

Plugging in these values, we have:

= (1)(cos θ) + (0)(sin θ)

= cos θ + 0

= cos θ

Therefore, we have proved that sin(90° + θ) equals cos θ.

I hope that helps! Let me know if you have any further questions.