Introduction to Trigonometric Identities and Problem Context

Trigonometric identities are fundamental in solving a wide range of mathematical problems within the realms of calculus, physics, and engineering. One such problem involves finding the value of sin x cos x given that cos x - sin x √2/5. This article delves into the process of solving such a problem, employing algebraic manipulation and trigonometric identities.

Solving for sin x cos x using Trigonometric Identities

Given the equation cos x - sin x √2/5, we aim to find the value of sin x cos x. To proceed, we will square both sides of the given equation to utilize the Pythagorean identity and derive the required value step by step.

Step-by-Step Solution

1. Square Both Sides of the Equation

By squaring both sides of the equation cos x - sin x √2/5, we get:

[(cos x - sin x)^2 left(frac{sqrt{2}}{5}right)^2]

Expanding the left-hand side using the identity (a - b)^2 a^2 b^2 - 2ab and simplifying the right-hand side, we obtain:

cos^2 x sin^2 x - 2sin x cos x 2/5

Since cos^2 x sin^2 x 1, the equation simplifies to:

[1 - 2sin x cos x frac{2}{5}]

2. Isolate and Solve for sin x cos x

To isolate sin x cos x, subtract 1 from both sides and multiply by -1:

-2sin x cos x 2/5 - 1

-2sin x cos x 2/5 - 5/5

-2sin x cos x -3/5

Divide both sides by -2:

sin x cos x 3/10

This result is derived directly from the algebraic manipulation and the use of the Pythagorean identity.

Alternative Methods and Verifications

There are multiple ways to solve this problem, and each method confirms the same result:

1. Using the Alternative Identity Approach

The expression can also be solved by considering the identity:

[frac{1}{sqrt{2}} cos x - frac{1}{sqrt{2}} sin x frac{1}{sqrt{5}}]

This can be rewritten as:

[sin (45 - x) frac{1}{sqrt{5}} sin 26.56°]

Thus, we have:

x 18.44°

And hence:

sin x cos x 0.3

2. Re-evaluating the Squaring Method

Given: cos x - sin x √2/5

Square Both Sides:

[cos^2 x sin^2 x - 2sin x cos x frac{2}{5}]

Simplify to:

1 - 2sin x cos x 2/5

2sin x cos x 1 - 2/5 3/5

sin x cos x 3/10

Conclusion and Importance of Understanding Trigonometric Identities

Understanding and applying trigonometric identities is crucial in solving various problems in mathematics and related fields. This example illustrates the power of algebraic manipulation and the use of fundamental identities to simplify complex expressions. The result we obtained, sin x cos x 0.3, can be useful in multiple contexts, including calculus, physics, and engineering.

Note: The multiple methods of solving this problem demonstrate the versatility of trigonometric identities and the importance of verifying solutions through different approaches.