Question

How do you simplify $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v $ ?

Answer 1

Hint: In this question, we are given a trigonometric expression and we need to simplify it. For this, we will first apply the formula of subtraction of angles in sine and cosine function $ \sin \left( x-y \right) $ is equal to $ \cos \left( x-y \right) $ and cos(x-y) is equal to $ \cos x\cos y+\sin x\sin y $ . After that we will cancel the terms with opposite signs. At last we will apply $ {{\sin }^{2}}x+{{\cos }^{2}}x=1 $ to simplify our expression.

Complete step by step answer:
Here we are given the expression as, $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v $ .
Let us separate the terms and simplify them one by one. We know our first term as $ \sin \left( u-v \right)\cos v $ .
As we can see, there is a difference in the angles in the sine function. So we can apply the formula of subtraction of angles in sine here given by $ \sin \left( x-y \right)=\sin \cos y-\cos x\sin y $ . Here x is equal to u and y is equal to v. So our term becomes $ \left( \sin u\cos v-\cos u\sin v \right)\cos v $ .
Multiplying cosv inside the bracket terms we get $ \sin u{{\cos }^{2}}v-\cos u\sin v\cos v $ .
We have our second term as, $ \cos \left( u-v \right)\sin v $ .
As we can see there is a difference in the angles in cosine function. So let us apply the formula here given by $ \cos \left( x-y \right)=\cos x\cos y+\sin x\sin y $ . Here x is equal to u and y is equal to v. So our term becomes $ \left( \cos u\cos v+\sin u\sin v \right)\sin v $ .
Multiplying sinv inside the bracket terms we get $ \cos u\cos v\sin v+\sin u{{\sin }^{2}}v $ .
Putting values of both these terms in the expression we get $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v=\sin u{{\cos }^{2}}v-\cos u\sin v\cos v+\cos u\sin v\cos v+\sin u{{\sin }^{2}}v $ .
Cancelling positive cosusinvcosv with the negative cosusinvcosv we get, $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v=\sin u{{\cos }^{2}}v+\sin u{{\sin }^{2}}v $ .
Taking sinu common from both the terms we have, $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v=\sin u\left( {{\cos }^{2}}v+{{\sin }^{2}}v \right) $ .
Angles of both sine squared terms and cosine square terms are the same. So we can apply the formula of $ {{\sin }^{2}}x+{{\cos }^{2}}x=1 $ here, we get $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v=\sin u\left( 1 \right)\Rightarrow \sin u $ .
Therefore $ \sin \left( u-v \right)\cos v+\cos \left( u-v \right)\sin v=\sin u $ .
Hence the required simplified expression is sinu.

Note:
 Students should take care of signs in the formula of addition and subtraction of angles in the sine and cosine functions. While applying $ {{\sin }^{2}}x+{{\cos }^{2}}x=1 $ make sure that the angle of both the terms is the same.