Mistake in Cos(A + B)

We do encounter question like,

"Find the angle of A in cos (A + 45) = 0.42 ".

What do you do?

Two solutions are presented as below:

Solution 1:
cos (A + 45) = 0.42
==> A + 45 = cos^-1 0.42
==> A = 65.17 - 45 = 20.17 (Answer)

Solution 2:
cosA + cos45 = 0.42
==> cosA = 0.42 - 0.707 = - 0.287
==> A = cos^-1(-0.287)
==> A = 106.69 (Answer)

You can see that the 2 answers are different.
Why? Or is there 2 valid answers?

Looking carefully at the solutions above, you will see two concepts in approaching the solving.

The first working went through the conventional inverse cosine operation using the summed up angle (A + 45) as a piece.

The second solution used the concept of algebraic factorising to split the angles A and 45 before processing them separately.
What is wrong here?

To reveal the answer in advance, the first solution is correct while the second has a common mathematical fault.

cos (A + 45) means an operation of cosine onto the angles (A + 45) as a whole.

"cos" is not a variable to be operated upon.
Therefore, "cos" cannot be factorised!

The step, cos (A + 45), cannot be equal to cosA + cos45.
This is a common mistake that need to be removed from the brain.
Press the "Delete" button.

With this post, your trigonometry is getting better right?
Cheers!

You may visit this post for more mistakes to be avoided.

;)