Proving Trigonometric Identities: Techniques and Proofs

Trigonometry plays a vital role in mathematics, particularly in simplifying and verifying complex trigonometric expressions. In this article, we will explore the techniques used to prove several key trigonometric identities, specifically focusing on verifying the identities involving sin(3x)sin(2x) - sin(x).

Proving sin(3x)sin(2x) - sin(x) 4sin(x)cos(3x/2)cos(x/2)

Step 1: Start with the left-hand side (LHS) of the equation.

LHS: sin(3x)sin(2x) - sin(x)

Step 2: Use the product-to-sum identities to simplify the expressions.

{sin(3x) - sin(x)}sin(2x)

Step 3: Apply the identity 2cos(frac{a b}{2})sin(frac{a-b}{2}) sin(a) - sin(b) to further simplify.

2cos(frac{3x x}{2})sin(frac{3x-x}{2})sin(2x)

Step 4: Simplify the expression inside the cosine and sine functions.

2cos(2x)sin(x)sin(2x)

Step 5: Use the double angle identity for sine, i.e., sin(2x) 2sin(x)cos(x).

2cos(2x)sin(x) * 2sin(x)cos(x)

Step 6: Simplify the product of sine and cosine functions.

4sin(x)cos(2x)cos(x)

Step 7: Use the double angle identity for cosine, i.e., cos(2x) 2cos^2(x) - 1).

4sin(x)(2cos^2(x) - 1)cos(x)

Step 8: Apply the identity 2cos(frac{3x}{2})cos(frac{x}{2}) 2cos^2(frac{3x}{2}) - 1.

4sin(x)2cos(frac{3x}{2})cos(frac{x}{2})

Step 9: Simplify the right-hand side (RHS) of the equation.

RHS: 4sin(x)cos(frac{3x}{2})cos(frac{x}{2})

Conclusion: Since both the LHS and RHS simplify to the same expression, the identity is proven.

Proving sin(3x)sin(2x) - sin(x) 3sin(x) - 4sin^3(x)

Step 1: Start with the left-hand side (LHS) of the equation.

LHS: sin(3x)sin(2x) - sin(x)

Step 2: Use the triple angle identity for sine, i.e., sin(3x) 3sin(x) - 4sin^3(x).

3sin(x) - 4sin^3(x) - sin(x)

Step 3: Simplify the expression by combining like terms.

3sin(x) - sin(x) - 4sin^3(x)

Step 4: Factor out sin(x) from the simplified expression.

2sin(x) - 4sin^3(x)

Step 5: Factor out 2sin(x) from the expression.

2sin(x)(1 - 2sin^2(x))

Step 6: Use the double angle identity for cosine, i.e., cos(2x) 1 - 2sin^2(x).

2sin(x)cos(2x)

Conclusion: Since the LHS simplifies to the RHS, the identity is proven.

Proving sin(3x)sin(2x) - sin(x) 2sin(5x/2)cos(x/2) - 2sin(x/2)cos(x/2)

Step 1: Start with the left-hand side (LHS) of the equation.

LHS: 2sin(3x)2x/2cos(3x) - 2x/2 - sin(x)

Step 2: Use the product-to-sum identities to simplify.

2sin(3x)2x/2cos(3x) - 2x/2 - sin(x)

Step 3: Use the identity sin(ab) 2sin(ab/2)cosa-b/2 to simplify further.

2sin(5x/2)cos(x/2) - 2sin(x/2)cos(x/2)

Conclusion: Since the LHS simplifies to the RHS, the identity is proven.

Conclusion

In summary, we have proven several important trigonometric identities involving sin(3x)sin(2x) - sin(x). These proofs rely on fundamental trigonometric identities such as product-to-sum identities, double angle identities, and triple angle identities. Understanding and practicing these techniques are crucial for simplifying and verifying complex trigonometric expressions.

By mastering these techniques, you can solve a wide range of problems in trigonometry and related fields, making these skills invaluable for students and professionals alike.