Rewrite The Equation To Simplify Or Make Sense.Solve The Equation:$\cos X - \cos^3 X = \tan X (\cos X - \cos X \sin^2 X$\]

Introduction

Trigonometric equations can be complex and challenging to solve, but with the right approach, they can be simplified and made more manageable. In this article, we will focus on rewriting the given equation to simplify or make sense. We will use trigonometric identities and formulas to manipulate the equation and arrive at a solution.

The Given Equation

The given equation is:

$$\cos x - \cos^3 x = \tan x (\cos x - \cos x \sin^2 x)$$

Step 1: Simplify the Right-Hand Side

To simplify the right-hand side of the equation, we can start by using the trigonometric identity:

$$\tan x = \frac{\sin x}{\cos x}$$

Substituting this into the equation, we get:

$$\cos x - \cos^3 x = \frac{\sin x}{\cos x} (\cos x - \cos x \sin^2 x)$$

Step 2: Simplify the Expression Inside the Parentheses

We can simplify the expression inside the parentheses by factoring out $\cos x$:

$$\cos x - \cos x \sin^2 x = \cos x (1 - \sin^2 x)$$

Using the trigonometric identity:

$$1 - \sin^2 x = \cos^2 x$$

We can rewrite the expression as:

$$\cos x - \cos x \sin^2 x = \cos x \cos^2 x$$

Step 3: Substitute the Simplified Expression Back into the Equation

Substituting the simplified expression back into the equation, we get:

$$\cos x - \cos^3 x = \frac{\sin x}{\cos x} (\cos x \cos^2 x)$$

Step 4: Simplify the Right-Hand Side Further

We can simplify the right-hand side further by canceling out the $\cos x$ term:

$$\cos x - \cos^3 x = \sin x \cos^2 x$$

Step 5: Factor Out $\cos x$ from the Left-Hand Side

We can factor out $\cos x$ from the left-hand side of the equation:

$$\cos x (1 - \cos^2 x) = \sin x \cos^2 x$$

Step 6: Simplify the Expression Inside the Parentheses

We can simplify the expression inside the parentheses by using the trigonometric identity:

$$1 - \cos^2 x = \sin^2 x$$

We can rewrite the expression as:

$$\cos x \sin^2 x = \sin x \cos^2 x$$

Step 7: Divide Both Sides by $\cos x$

We can divide both sides of the equation by $\cos x$ to get:

$$\sin^2 x = \tan x \cos x$$

Step 8: Simplify the Right-Hand Side

We can simplify the right-hand side by using the trigonometric identity:

$$\tan x = \frac{\sin x}{\cos x}$$

Substituting this into the equation, we get:

$$\sin^2 x = \frac{\sin x}{\cos x} \cos x$$

Step 9: Cancel Out the $\cos x$ Term

We can cancel out the $\cos x$ term on both sides of the equation:

$$\sin^2 x = \sin x$$

Step 10: Solve for $\sin x$

We can solve for $\sin x$ by dividing both sides of the equation by $\sin x$:

$$\sin x = 1$$

Conclusion

In this article, we have simplified the given trigonometric equation by using various trigonometric identities and formulas. We have arrived at the solution $\sin x = 1$, which is a valid solution for the given equation.

Final Answer

The final answer is $\boxed{1}$.

Note

Q: What is the given equation?

A: The given equation is $\cos x - \cos^3 x = \tan x (\cos x - \cos x \sin^2 x)$.

Q: How do I simplify the right-hand side of the equation?

A: To simplify the right-hand side of the equation, you can start by using the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$. Substituting this into the equation, you get $\cos x - \cos^3 x = \frac{\sin x}{\cos x} (\cos x - \cos x \sin^2 x)$.

Q: How do I simplify the expression inside the parentheses?

A: You can simplify the expression inside the parentheses by factoring out $\cos x$. This gives you $\cos x - \cos x \sin^2 x = \cos x (1 - \sin^2 x)$. Using the trigonometric identity $1 - \sin^2 x = \cos^2 x$, you can rewrite the expression as $\cos x - \cos x \sin^2 x = \cos x \cos^2 x$.

Q: How do I substitute the simplified expression back into the equation?

A: You can substitute the simplified expression back into the equation by replacing the original expression with the simplified one. This gives you $\cos x - \cos^3 x = \frac{\sin x}{\cos x} (\cos x \cos^2 x)$.

Q: How do I simplify the right-hand side further?

A: You can simplify the right-hand side further by canceling out the $\cos x$ term. This gives you $\cos x - \cos^3 x = \sin x \cos^2 x$.

Q: How do I factor out $\cos x$ from the left-hand side?

A: You can factor out $\cos x$ from the left-hand side of the equation by grouping the terms. This gives you $\cos x (1 - \cos^2 x) = \sin x \cos^2 x$.

Q: How do I simplify the expression inside the parentheses?

A: You can simplify the expression inside the parentheses by using the trigonometric identity $1 - \cos^2 x = \sin^2 x$. This gives you $\cos x \sin^2 x = \sin x \cos^2 x$.

Q: How do I divide both sides by $\cos x$?

A: You can divide both sides of the equation by $\cos x$ to get $\sin^2 x = \tan x \cos x$.

Q: How do I simplify the right-hand side?

A: You can simplify the right-hand side by using the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$. Substituting this into the equation, you get $\sin^2 x = \frac{\sin x}{\cos x} \cos x$.

Q: How do I cancel out the $\cos x$ term?

A: You can cancel out the $\cos x$ term on both sides of the equation by dividing both sides by $\cos x$. This gives you $\sin^2 x = \sin x$.

Q: How do I solve for $\sin x$?

A: You can solve for $\sin x$ by dividing both sides of the equation by $\sin x$. This gives you $\sin x = 1$.

Q: What is the final answer?

A: The final answer is $\boxed{1}$.

Q: What is the given equation?

A: The given equation is $\cos x - \cos^3 x = \tan x (\cos x - \cos x \sin^2 x)$.

Q: How do I simplify the equation?

A: You can simplify the equation by using various trigonometric identities and formulas. The steps to simplify the equation are:

1. Simplify the right-hand side of the equation.
2. Simplify the expression inside the parentheses.
3. Substitute the simplified expression back into the equation.
4. Simplify the right-hand side further.
5. Factor out $\cos x$ from the left-hand side.
6. Simplify the expression inside the parentheses.
7. Divide both sides by $\cos x$.
8. Simplify the right-hand side.
9. Cancel out the $\cos x$ term.
10. Solve for $\sin x$.

Q: What is the solution to the equation?

A: The solution to the equation is $\sin x = 1$.

Q: What is the final answer?

A: The final answer is $\boxed{1}$.