# Verify the identity cos29 * cos1 - sin29 * sin1 = cos 28 * cos 2 + sin 28 * sin 2.

### 3 Answers | Add Yours

To verify

cos29 * cos1 - sin29 * sin1 = cos 28 * cos 2 + sin 28 * sin 2.

Solution:

We know that cosacosb - sinasinb = cos(a+b). Applying this to Left side of the equation, we get :

cos29cos 1 -sin28sin1 = cos(29+1) = cos30.

Aplying the rule to the right side of the given equation, we get:

cos28 cos2- sin28sin2 = cos(28+2 ) = cos30.

So the given identity holds true.

Here we have to verify that

cos29 * cos1 - sin29 * sin1 = cos 28 * cos 2 + sin 28 * sin 2.

Now we know that cos (a - b) = cos a*cos b + sin a*sin b

and cos (a + b) = cos a*cos b - sin a*sin b

In cos 29 * cos 1 - sin 29 * sin 1, a = 29 and b =1

Therefore cos 29 * cos 1 - sin 29 * sin 1 = cos ( 29 +1)= cos 30.

In cos 28 * cos 2 + sin 28 * sin 2, a= 28 and b=2 but this expression is of cos(a - b) => cos (28 - 2 ) = cos (26).

The sign between cos 28 * cos 2 and sin 28 * sin 2 should have been** minus instead of plus** for both sides to equal cos 30.

**In the present form cos 30 is not equal to cos 26.**

We notice that the given sum is the expanding of the followings:

cos (a - b) = cos a*cos b + sin a*sin b

and

cos (a + b) = cos a*cos b - sin a*sin b

If we'll substitute a and b by 28 and 2, we'll get the given sum, that has to be calculated:

cos (28-2) = cos 28 * cos 2 + sin 28 * sin 2

**cos 28 * cos 2 + sin 28 * sin 2 = cos 26**

Now, we'll substitute a and b by 29 and 1, to calculate the difference:

cos (29+1) = cos29 * cos1 - sin29 * sin1

**cos29 * cos1 - sin29 * sin1 = cos 30**

It is obvious that the result of the difference is not the same with the result of the sum:

**cos 30 is not equal to cos 26!**

Note: In case of identity between the given expression we'll have to make the correction in the expression cos 28 * cos 2 + sin 28 * sin 2.

This expression has to be a difference instead of a sum:

cos 28 * cos 2 - sin 28 * sin 2 = cos (28+2) = cos 30