Let's solve the equation \(\frac{\cos A - \sin A}{\cos A + \sin A} = \sec(2A) - \tan(2A)\) step by step.
First, consider the left-hand side of the equation:
[tex]\[
\text{Left-hand side: } \frac{\cos A - \sin A}{\cos A + \sin A}
\][/tex]
Next, let's consider the right-hand side of the equation:
[tex]\[
\text{Right-hand side: } \sec(2A) - \tan(2A)
\][/tex]
Now let's break it down.
### Simplifying the Left-Hand Side
The left-hand side is already in a simplified form involving trigonometric functions:
[tex]\[
\frac{\cos A - \sin A}{\cos A + \sin A}
\][/tex]
### Simplifying the Right-Hand Side
Evaluate the right-hand side using known trigonometric identities:
[tex]\[
\sec(2A) = \frac{1}{\cos(2A)}, \quad \tan(2A) = \frac{\sin(2A)}{\cos(2A)}
\][/tex]
Thus, the right-hand side can be written as:
[tex]\[
\sec(2A) - \tan(2A) = \frac{1}{\cos(2A)} - \frac{\sin(2A)}{\cos(2A)} = \frac{1 - \sin(2A)}{\cos(2A)}
\][/tex]
### Comparing Both Sides
To check the equality, let's compare:
[tex]\[
\frac{\cos A - \sin A}{\cos A + \sin A} \quad \text{and} \quad \frac{1 - \sin(2A)}{\cos(2A)}
\][/tex]
Upon simplifying expressions and calculations, we find that:
[tex]\[
\frac{\cos A - \sin A}{\cos A + \sin A} = \frac{1}{\tan(A + \frac{\pi}{4})}
\][/tex]
Whereas:
[tex]\[
\sec(2A) - \tan(2A) = -\tan(2A) + \sec(2A)
\][/tex]
### Conclusion
Upon simplification, it is evident that the two expressions:
[tex]\[
\frac{\cos A - \sin A}{\cos A + \sin A} \quad \text{and} \quad \sec(2A) - \tan(2A)
\][/tex]
do not simplify to the same value. Hence, the given equation:
[tex]\[
\frac{\cos A - \sin A}{\cos A + \sin A} = \sec(2A) - \tan(2A)
\][/tex]
is not true.
