Prepared: Thu May 20 15:12:07 CDT 1999

MATH 1151

QUIZ VI

1) Prove the following identity:

We observe that the left hand side is more complicated than the other side, so we start with the left hand side and try to show that it equals right hand side. Observe also that right hand side is in terms of only, so we'd better rewrite in terms of . For this purpose, recall the first Pythagorean identity (Pg 341):

Using the identity with replaced by and solving for we obtain:

Now using this in the left hand side we get what we were looking for:

2) Use the addition-subtraction identities to verify

Recall the addition identity for sin (Pg 351):

Using the above identity for , we get

is a special angle, therefore you should recall immediately that and . Even if you cannot recall these values, you are allowed to use your calculator to find them. Using those values back in gives what we want:

3) a) Write the product as a sum by using the product-to-sum identitites:

Recall the product-to-sum identity for a product of two cosines (Pg 364):

Use this identity with replaced by and replaced by :

The above answer is an acceptable answer, but is not quite simplified. This is not easy to see, so I would not expect the following steps from you. If you have done the following, perfect. If you have done only the above part, that's fine also.

Observe that since cos is an even function we can say

One further step will be to use the addition identity with cosine (which I will not recall but I will refer you to Addition-Subtraction identities on pg 351):

which can be seen by evaluating the values of and . So the answer is

in the most simplified form.

b) Write the sum as a product by using the sum-to-product identitites:

Recall the sum-to-product identity for the difference of two sines (Pg 365):

Using this identity with replaced by 4y and replaced by 3y: