Cos²A/1-tan²A +sin³A/sinA-cosA =1+sinA.cosA

Let's solve the equation step by step to verify if the given trigonometric identity is correct.

The given equation is:

\[
\frac{\cos^2A}{1 - \tan^2A} + \frac{\sin^3A}{\sinA - \cosA} = 1 + \sinA \cdot \cosA
\]

### Step 1: Simplify the first term \(\frac{\cos^2A}{1 - \tan^2A}\)
We know that \(\tanA = \frac{\sinA}{\cosA}\), so \(\tan^2A = \frac{\sin^2A}{\cos^2A}\).

Substituting \(\tan^2A\) into the equation:

\[
\frac{\cos^2A}{1 - \frac{\sin^2A}{\cos^2A}} = \frac{\cos^2A}{\frac{\cos^2A - \sin^2A}{\cos^2A}} = \frac{\cos^2A \cdot \cos^2A}{\cos^2A - \sin^2A} = \frac{\cos^4A}{\cos^2A - \sin^2A}
\]

### Step 2: Simplify the second term \(\frac{\sin^3A}{\sinA - \cosA}\)

\[
\frac{\sin^3A}{\sinA - \cosA}
\]

This expression remains as is since it cannot be simplified further.

### Step 3: Combine the expressions and compare with the right side

So far, the left-hand side (LHS) is:

\[
\frac{\cos^4A}{\cos^2A - \sin^2A} + \frac{\sin^3A}{\sinA - \cosA}
\]

The right-hand side (RHS) is:

\[
1 + \sinA \cdot \cosA
\]

### Step 4: Analyze the expression

It is apparent that these terms might not be algebraically equal. To confirm, let's consider specific values for \(A\).

For example, let's consider \(A = 45^\circ\) (where \(\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}\)):

- LHS: \( \frac{(\frac{1}{\sqrt{2}})^4}{(\frac{1}{\sqrt{2}})^2 - (\frac{1}{\sqrt{2}})^2} + \frac{(\frac{1}{\sqrt{2}})^3}{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}} \)
- RHS: \(1 + \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = 1 + \frac{1}{2}\)

On substitution, you'll find that the expressions on LHS and RHS do not match. This implies that the given equation is **not an identity**.

If you need further steps for verification, simplifications, or specific questions related to trigonometric identities, please let me know!

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Do you want more details or have any questions?

Here are some related questions to explore further:

1. How do you derive trigonometric identities?
2. Can the equation be an identity for any specific value of \(A\)?
3. What are common methods for verifying trigonometric identities?
4. How does the Pythagorean identity relate to trigonometric equations?
5. What is the significance of \(\tan^2 A\) in trigonometric identities?
6. Can this identity hold under specific transformations?
7. How does angle substitution affect trigonometric expressions?
8. What are the common mistakes when simplifying trigonometric identities?

**Tip:** When verifying trigonometric identities, substituting specific angles can sometimes help to quickly check if the identity might be valid.

Explore how to verify the trigonometric identity Cos²A/1-tan²A + sin³A/sinA-cosA = 1+sinA.cosA with a step-by-step solution. Understand the application of trigonometric identities and specific values of A to determine the validity of the equation. Suitable for high school students and anyone interested in trigonometry.

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