Matzic|

This is most useful mathematical formula in trigonometry. Actually, this formula tells relationship between two multiple angles. In this trigonometric function C and D are two multiple angles of a right angle triangle. Now, our main objective is two simplify summation of two similar trigonometric functions with two different multiple angles, which means we have to simplify below trigonometric function mathematically.

SinC + SinD

It is not easy to find the solution for this trigonometric equation directly, which means we have to approach in different way in order to find the required solution.

Let us assume C = A + B and D = A – B

In the assumption, we are considering that the summation and subtraction of two different angles give multiple angles C and D. Now submit C and D value in terms of A and B angles.

SinC + SinD = Sin(A + B) + Sin(A – B)

We have already proved that

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

And

Sin(A – B) = SinA.CosB – CosA. SinB

Now, substitute the expansion of each formula in above trigonometric function to get required solution.

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

= 2.SinA.CosB

We get simplified equation in terms of A and B. Now we have to convert this equation in terms of C and D. This

If C and D equal to A + B and A – B then we can prove that A = (C + D)/2 and B = (C – D)/2.

2.SinA.CosB = 2.Sin[(C + D)/2].Cos[(C - D)/2]

We have successfully proved this Sine function in terms of Sine and Cosine.

SinC + SinD = 2.Sin[(C + D)/2].Cos[(C - D)/2]

Conclusion:

SinC + SinD = 2.Sin[(C + D)/2].Cos[(C - D)/2]