\left(\frac{1-tan\:x}{sec\:x}\right)^2+\:2\:sen\:x\:cos\:x\:=1

The given equation is:

\[
\left(\frac{1 - \tan x}{\sec x}\right)^2 + 2\sin x \cos x = 1
\]

Let's work through this step by step.

### Step 1: Simplify \(\frac{1 - \tan x}{\sec x}\)

We know that:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\sec x = \frac{1}{\cos x}\)

Substitute these into the expression:

\[
\frac{1 - \tan x}{\sec x} = \frac{1 - \frac{\sin x}{\cos x}}{\frac{1}{\cos x}}
\]

Simplifying the numerator:

\[
1 - \frac{\sin x}{\cos x} = \frac{\cos x - \sin x}{\cos x}
\]

Now the whole expression becomes:

\[
\frac{\frac{\cos x - \sin x}{\cos x}}{\frac{1}{\cos x}} = (\cos x - \sin x)
\]

### Step 2: Substitute and simplify the equation

Now substitute \(\cos x - \sin x\) into the equation:

\[
(\cos x - \sin x)^2 + 2\sin x \cos x = 1
\]

### Step 3: Expand \((\cos x - \sin x)^2\)

Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), we expand:

\[
(\cos x - \sin x)^2 = \cos^2 x - 2\sin x \cos x + \sin^2 x
\]

Thus the equation becomes:

\[
\cos^2 x - 2\sin x \cos x + \sin^2 x + 2\sin x \cos x = 1
\]

### Step 4: Simplify further

Notice that the terms \(-2\sin x \cos x\) and \(+2\sin x \cos x\) cancel each other out:

\[
\cos^2 x + \sin^2 x = 1
\]

### Step 5: Use the Pythagorean identity

We know from the Pythagorean identity that:

\[
\cos^2 x + \sin^2 x = 1
\]

Therefore, the equation simplifies to:

\[
1 = 1
\]

This is a true statement, meaning the original equation is an identity and holds for all values of \(x\).

### Conclusion

The equation is an identity and is true for all values of \(x\).

---

Would you like more details on any of the steps? Here are some related questions for further exploration:

1. What are the key trigonometric identities used in this solution?
2. How can the Pythagorean identity be derived geometrically?
3. What are the other forms of simplifying expressions involving \(\sec x\) and \(\tan x\)?
4. Can this problem be approached using a graphical method?
5. How would this equation change if it were in terms of radians?

**Tip:** Understanding how trigonometric identities relate to each other can greatly simplify solving more complex problems.

\left(\frac{1 - \tan x}{\sec x}\right)^2 + 2\sin x \cos x = 1

Explore the step-by-step solution for simplifying the trigonometric equation \left(\frac{1 - \tan x}{\sec x}\right)^2 + 2\sin x \cos x = 1 using key trigonometric identities such as the Pythagorean identity. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation