How to Prove a Trigonometric Identity

[chenny1]

cos = sin (pi/2-theta)

 

[Mark44]

What have you tried? You need to show an attempt at a solution before anyone can help you out. What trig identities do you know?

 

[chenny1]

Wait. I'm just asking one question.
The trig identity I was trying to use was cos = sin (pi/2-theta).
I'm sorry, I haven't taken trig for along time and I'm just trying to brush up for my admissions test. I'm not too positive what it is asking when it states establish an identity. Am I suppose to prove it?

 

[Mark44]

1. Establish the identity sin(3pi/2-theta) = - cos theta

2. cos = sin (pi/2-theta)

3. sin (3pi/2-theta = - sin (pi/2-theta)

Establishing an identity means proving that it is a true statement for all values of the variable. Your numbered statements threw me off. I thought you were asking how to prove three statements.

To prove an identity, you genearlly want to start with one side and use identities to manipulate it, showing that it is the same as the other side.

Work with sin(3pi/2 - theta), using the identity for sine of the difference of two angles. You will also need to know the values of the sine and cosine functions at a specific angle.

Be sure to include the angle, and be consistent in your use of parentheses. In #2 you cos, which is just the name of the function. In #3, you're missing a right parenthesis.

Here's an example of proving an identity.

Show that (x + 2)² - (x - 2)² = 8x, for all x.

It's usually a good idea to start with the side that seems most complicated.
(x + 2)² - (x - 2)² = x² + 4x + 4 - (x² - 4x + 4) = x² + 4x + 4 - x² + 4x - 4 = 4x + 4x = 8x.

This shows that (x + 2)² - (x - 2)² = 8x. Since there are no restrictions on x in any step, this is an identity that is true for all values of x.