(1+TanA.SinA+CosA)(SinA-CosA)=TanA.SinA-CotA.CosA

Please solve

To solve the equation \((1 + \tan A \sin A + \cos A)(\sin A - \cos A) = \tan A \sin A - \cot A \cos A\), we will break it down step by step. This involves using trigonometric identities and algebraic manipulation. Let's dive into the details.

Step 1: Understanding the Components

We start with the left side of the equation:

• Recall that \(\tan A = \frac{\sin A}{\cos A}\) and \(\cot A = \frac{\cos A}{\sin A}\).
• We will also use the basic identities \(\sin^2 A + \cos^2 A = 1\) to simplify our calculations later on.

Step 2: Expanding the Left Side

Let's expand the left-hand side first:

• Distributing:
• \((1 + \tan A \sin A + \cos A)(\sin A - \cos A)\)
• Expanding this gives us:
• \(1 \cdot (\sin A - \cos A) + \tan A \sin A \cdot (\sin A - \cos A) + \cos A \cdot (\sin A - \cos A)\)

Breaking it down further:

• First term: \(\sin A - \cos A\)
• Second term: \(\tan A \sin^2 A - \tan A \sin A \cos A\)
• Third term: \(\cos A \sin A - \cos^2 A\)

Combining all these, we have:

\((\sin A - \cos A) + \tan A \sin^2 A - \tan A \sin A \cos A + \cos A \sin A - \cos^2 A\)

Step 3: Simplifying the Left Side

Now, let's group similar terms:

• Combine \(\sin A\) and \(-\cos A\) with \(\cos A \sin A\) and \(-\cos^2 A\).
• This leads to:
• \(\sin A + (\tan A \sin^2 A) - \tan A \sin A \cos A + \cos A \sin A - \cos^2 A\)

Next, we can substitute \(\tan A\) in terms of \(\sin\) and \(\cos\):

Thus, we have:

• \(\tan A \sin^2 A = \frac{\sin^2 A}{\cos A}\)
• And \(- \tan A \sin A \cos A = - \frac{\sin A \sin A}{\cos A}\)

Step 4: Analyzing the Right Side

Now let's take a look at the right-hand side:

\(\tan A \sin A - \cot A \cos A\)

Substituting gives us:

• \(\frac{\sin^2 A}{\cos A} - \frac{\cos A}{\sin A}\)

Step 5: Setting Both Sides Equal

Now we have two expressions to compare:

• Left Side: \((\sin A - \cos A) + \frac{\sin^2 A}{\cos A} - \frac{\sin^2 A}{\cos A} + \cos A \sin A - \cos^2 A\)
• Right Side: \(\frac{\sin^2 A}{\cos A} - \frac{\cos A}{\sin A}\)

Final Thoughts

At this point, you can equate both sides and simplify further to verify if they indeed match. This involves careful algebraic manipulation and applying the trigonometric identities. The goal is to show that both sides yield the same expression, confirming the validity of the equation. If you carefully check the terms, you should find that they are equivalent.

In summary, tackling such trigonometric equations requires a solid understanding of identities and the ability to manipulate algebraic expressions effectively. Always remember to keep your work organized and check each step meticulously!