Solving Trigonometric Equations: Finding the Value of an Expression

This problem asks us to find the value of a specific trigonometric expression, $\text{Cos}^8 x + 2 \text{Cos}^6 x + \text{Cos}^4 x$, given a trigonometric equation, $\text{Sin } x + \text{Sin}^2 x = 1$. To solve this, we need to manipulate the given equation and use trigonometric identities to simplify the expression.

Step-by-Step Solution

Let's start with the given equation:

$\text{Sin } x + \text{Sin}^2 x = 1$

We can rearrange this equation to isolate $\text{Sin } x$:

$\text{Sin } x = 1 - \text{Sin}^2 x$

Recall the fundamental trigonometric identity: $\text{Sin}^2 x + \text{Cos}^2 x = 1$. From this, we know that $1 - \text{Sin}^2 x = \text{Cos}^2 x$. Substituting this into our rearranged equation gives us a crucial relationship:

$\text{Sin } x = \text{Cos}^2 x$

This relationship is the key to solving the problem. Now, let's look at the expression we need to evaluate:

$\text{Cos}^8 x + 2 \text{Cos}^6 x + \text{Cos}^4 x$

We can rewrite each term in this expression using $\text{Cos}^2 x$:

• $\text{Cos}^8 x = (\text{Cos}^2 x)^4$
• $\text{Cos}^6 x = (\text{Cos}^2 x)^3$
• $\text{Cos}^4 x = (\text{Cos}^2 x)^2$

Now, substitute the relationship $\text{Cos}^2 x = \text{Sin } x$ into the expression:

$(\text{Sin } x)^4 + 2 (\text{Sin } x)^3 + (\text{Sin } x)^2$

This simplifies to:

$\text{Sin}^4 x + 2 \text{Sin}^3 x + \text{Sin}^2 x$

Let's examine this expression carefully. It looks like a perfect square trinomial. Recall the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. We can rewrite the expression as:

$(\text{Sin}^2 x)^2 + 2 (\text{Sin}^2 x)(\text{Sin } x) + (\text{Sin } x)^2$

Here, let $a = \text{Sin}^2 x$ and $b = \text{Sin } x$. The expression matches the form $a^2 + 2ab + b^2$, so it can be factored as $(a+b)^2$:

$(\text{Sin}^2 x + \text{Sin } x)^2$

Finally, we can use the original given equation, $\text{Sin } x + \text{Sin}^2 x = 1$, to find the value of this expression:

$(\text{Sin}^2 x + \text{Sin } x)^2 = (1)^2 = 1$

Thus, the value of $\text{Cos}^8 x + 2 \text{Cos}^6 x + \text{Cos}^4 x$ is 1.

Summary of Steps

1. Start with the given equation $\text{Sin } x + \text{Sin}^2 x = 1$.
2. Rearrange the equation to get $\text{Sin } x = 1 - \text{Sin}^2 x$.
3. Use the identity $\text{Sin}^2 x + \text{Cos}^2 x = 1$ to show $\text{Sin } x = \text{Cos}^2 x$.
4. Rewrite the expression $\text{Cos}^8 x + 2 \text{Cos}^6 x + \text{Cos}^4 x$ in terms of $\text{Cos}^2 x$.
5. Substitute $\text{Cos}^2 x = \text{Sin } x$ into the expression.
6. Recognize the resulting expression as a perfect square: $(\text{Sin}^2 x + \text{Sin } x)^2$.
7. Substitute the value from the original equation $\text{Sin } x + \text{Sin}^2 x = 1$ into the simplified expression.
8. Calculate the final value.

Revision Table: Key Identities

Additional Information: Understanding Trigonometric Relationships

Trigonometric problems often require skillful manipulation of equations and identities. The key insight in this problem was deriving the relationship $\text{Sin } x = \text{Cos}^2 x$ from the given equation. This allowed us to transform an expression involving powers of cosine into an expression involving powers of sine, which then conveniently simplified using the original equation. Recognizing algebraic patterns, like the perfect square trinomial here, is also a common technique in solving trigonometric problems.

This problem demonstrates how an equation relating sine and cosine of the same angle can simplify complex expressions. Always look for ways to use fundamental identities like $\text{Sin}^2 x + \text{Cos}^2 x = 1$ to establish connections between different trigonometric functions in the given problem.