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Chapter 5: Problem 98

Verify that \(2 \sin x \cos x=\sin 2 x\) by using a product-to-sum identity.

Short Answer

Expert verified
Yes, the two expressions \(2\sin(x)\cos(x)\) and \(\sin(2x)\) are indeed equivalent, as shown by the double-angle identity. This completes the verification.

Step by step solution

01

Identify Relevant Trigonometric Identity

One could use the double-angle identity which states that \(\sin(2x) = 2\sin(x)\cos(x)\). This is the tool to use here. It allows to write the sinus of a doubled angle in terms of sine and cosine of the original angle.
02

Use the Double-Angle Identity to Reformulate the Expression

With the double-angle identity, write the expression \(\sin(2x)\) as equivalent to \(2\sin(x)\cos(x)\).
03

Compare the Given Expression

Compare the expression gained on the right-hand side with the expression given in the exercise. The expressions \(2\sin(x)\cos(x)\) and \(\sin(2x)\) are equivalent as per the double-angle identity.

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