Trigonometric Identities Proof

In summary, using the given equalities, we can derive the following trigonometric identities: a. absolute value of cos x/2 = \sqrt{}1 + cosx/2, and b. absolute value of sin x/2 = \sqrt{}1 - cosx/2. To do this, we can set A=B in the identity cos(A+B) = cosAcosB - sinAsinB, and replace A with x/2 to get the desired equations. Additionally, we need to remove the minus sign from cos(-x) = -cosx, as it should be equal to cos x without the minus on the right side.
  • #1
whitehorsey
192
0
1. (A) sin(-x) = - sin x (C) cos(x+y) = cosxcosy - sinxsiny
(B) cos(-x) = - cos x (D) sin(x+y) = sinxcosy + cosxsiny
Use these equalities to derive the following trigonometric identities.
a. absolute value of cos x/2 = [tex]\sqrt{}1 + cosx/2[/tex]
b. absolute value of sin x/2 = [tex]\sqrt{}1 - cosx/2[/tex]



2.above



3. I'm stuck on these two and tried to think of different ways to solve it but I can't seem to get find a solution to it. Can you please explain how to derive those two equations? Thank You!
 
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  • #2
cos(A+B) = cosAcosB - sinAsinB, try setting A=B and then see what happens. Since there is an x/2, maybe you should replace A with that.
 
  • #3
(B) cos(-x) = - cos x
needs to be cos(-x) = cos x without the minus in front on the right side.
 

1. What are trigonometric identities?

Trigonometric identities are mathematical equations that relate different trigonometric functions to each other. They are useful in solving trigonometric equations and simplifying complex expressions.

2. Why is it important to prove trigonometric identities?

Proving trigonometric identities is important because it helps to verify the accuracy of the equations and provides a deeper understanding of the relationships between trigonometric functions. It also allows for the manipulation and simplification of trigonometric expressions, making them easier to work with in various mathematical applications.

3. What are the basic trigonometric identities?

The basic trigonometric identities include the Pythagorean identities, reciprocal identities, quotient identities, co-function identities, and even-odd identities. These identities can be used to prove more complex trigonometric identities.

4. What steps are involved in proving a trigonometric identity?

The steps for proving a trigonometric identity may vary, but generally, it involves using algebraic and trigonometric manipulation, substituting known identities, and simplifying both sides of the equation until they are equal.

5. What are some common techniques for proving trigonometric identities?

Some common techniques for proving trigonometric identities include using the double angle or half angle formulas, converting trigonometric functions to their exponential or logarithmic form, and using trigonometric identities in combination with algebraic manipulation to simplify expressions.

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