Question 1

Understanding the Problem

Accepted Answer

We are asked to show that the product of sine and cosine functions can be expressed as a sine function of a multiple angle. This is a fundamental trigonometric identity.

Question 2

Solving the Problem
Step 1: Recall the Product-to-Sum Identity

Accepted Answer

A key trigonometric identity is the product-to-sum formula for sine and cosine:

Question 3

Step 2: Apply the Identity

Accepted Answer

In our case, ## A = x ## and ## B = x ##. Substituting these values into the product-to-sum formula gives:

Question 4

Step 3: Simplify the Arguments

Accepted Answer

Simplifying the arguments of the sine functions yields:

Question 5

Step 4: Combine the Results

Accepted Answer

Substituting the simplified results back into the equation gives:

Question 6

Step 5: Final Simplification

Accepted Answer

Since ## \sin(0) = 0 ##, the equation simplifies to:

Question 7

Final Solution

Accepted Answer

Therefore, the derived identity is:

Deriving the Sine Cosine Identity: sin(x)cos(x) = (1/2)sin(2x)

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