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This is most useful trigonometric formula. We have already derived similar mathematical fundamental but it is in Sine terms. In the same way, we can also derive Cosine function with multiple angles. A, B and C are multiple angles of a right angle triangle.

Deriving the entire mathematical function is little bit complicated. However, we can reduce the difficulty by assuming two angles as one angle, which means let us assume an imaginary angle θ. It is nothing but summation or resultant angle of two angles. Θ = A + B. As per this assumption, we can write our Cosine function as stated below.

Cos(A + B + C) = Cos(θ + C)

We have already derived a two angle Cosine formula. Now, we have to apply that formula in order to further simply of this function.

Cos(A + B) = CosA.CosB – SinA.SinB

By using this mathematical formula, we are going to write our actual formula easily, which means

Cos(θ + C) = Cosθ.CosC – Sinθ.SinC

Now, resubmit the actual value of imaginary or resultant angle (θ) value in above trigonometric function.

Cos(A + B + C) = Cos(A + B).CosC – Sin(A + B).SinC

We almost simplified this trigonometric function but now we have to use another mathematical application, which is also already derived in previous article.

Sin(A + B) = SinA.CosB + CosA.SinB

Substitute these two trigonometric functions in our actual trigonometric function in order to get final simplification.

Cos(A + B + C) = [CosA.CosB – SinA.SinB].CosC – [SinA.CosB + CosA.SinB].SinC

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

Mathematically, we can represent the entire trigonometric function with Sigma Σ summation, which tells that summation of three terms.

Cos(A + B + C) = CosA.CosB.CosC – ΣSinA.SinB.CosC

This formula may not useful in entire trigonometry but it will definitely helpful us in dealing three term multiple angles and developing advanced trigonometry fundamentals.