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Developing a mathematical fundamental of Cosine with two different angles into its sub multiple angles help us to solve some complicated problems in trigonometry. Actually, these types of functions are basic steps to develop advanced trigonometric formulas. In other words, this trigonometric formula helped us in developing more advanced formulas, saving time and solving difficult problems easily in few steps.

We have also derived similar mathematical fundamental but it is in Sine function. Now, we are going to learn the basic procedure to express this formula into its sub multiple angles. The basic Cosine function’s two imaginary angles are C and D of a right angle triangle and the respective function would be like this.

CosC + CosD

It is not easy to simplify this function mathematically by using mathematical concepts. So, we have to use indirect approach in order to solve this function easily. For this, we have to assume two more angles A and B, which are part of the same right angle triangle. The relationship will be like this.

C = A + B and D = A – B, which means A = (C + D)/2 and B = (C – D)/2.

Now, convert the entire trigonometric function in terms of A and B for our convenience.

CosC + CosD = Cos(A + B) + Cos(A – B)

We have already proved previously that

Cos(A + B) = CosA.CosB – SinA.SinB

And

Cos(A – B) = CosA.CosB + SinA.SinB

Apply these two trigonometric fundamentals in order to simplify our actual function.

Cos(A + B) + Cos(A – B) = CosA.CosB – SinA.SinB + CosA.CosB + SinA.SinB

= 2.CosA.CosB

As per our assumption, A = (C + D)/2 and B = (C – D)/2.

Convert the entire solution into C and D terms to get actual solution.

CosC + CosD = 2.Cos[(C + D)/2].Cos[(C – D)/2]

Conclusion:

CosC + CosD = 2.Cos[(C + D)/2].Cos[(C – D)/2]