Introduction

In this article, we will explore a trigonometric problem that requires the application of algebraic manipulation to prove a specific identity. Given the equation cos θ sin θ √2 cos θ, we will demonstrate that cos θ - sin θ √2 sin θ. This proof involves a series of algebraic steps and the use of trigonometric identities. Let's begin by breaking down the problem and solving it step by step.

Step 1: Starting with the Given Equation

The given equation is:

cos θ sin θ √2 cos θ

Step 2: Rearranging the Equation

First, let's isolate sin θ on one side of the equation:

sin θ √2 cos θ - cos θ

Factor out cos θ on the right side:

sin θ cos θ (√2 - 1)

Step 3: Expressing cos θ in Terms of sin θ

Select cos θ x and sin θ y, and use the earlier result to express cos θ in terms of sin θ:

cos θ y / (√2 - 1)

Step 4: Substituting and Simplifying

Now, substitute cos θ in the expression cos θ - sin θ with the expression we derived:

cos θ - sin θ y / (√2 - 1) - y

Combine like terms:

cos θ - sin θ y [1 / (√2 - 1) - 1]

Step 5: Simplifying the Fractional Expression

Let's simplify the fraction 1 / (√2 - 1):

1 / (√2 - 1) (2 - 1) / (2 - 1) 1

Substitute back into the expression for cos θ - sin θ:

cos θ - sin θ y (1 - 1) √2 y

Since y sin θ, the final expression simplifies to:

cos θ - sin θ √2 sin θ

Conclusion

We have proven that cos θ - sin θ √2 sin θ given that cos θ sin θ √2 cos θ. This proof demonstrates the power of algebraic manipulation techniques in solving trigonometric equations and understanding the relationships between trigonometric functions.

Additional Resources

For more in-depth understanding of trigonometric identities and solving trigonometric equations, consider exploring additional online resources and textbooks such as:

Trigonometric Identities (MathIsFun) Khan Academy Trigonometry Trigonometric Identities (Paul's Online Math Notes)