Question

If \(\cos \theta + \cos^2 \theta = 1\) then the value of \(\sin^2 \theta + \sin^4 \theta\) is

• A. -1
• B. 1
• C. 0
• D. 2

Hint:

The key to this type of problem is to use the given equation to find a substitution. Rearranging the given equation using the Pythagorean identity is the crucial first step.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires manipulating a given trigonometric equation to find the value of another trigonometric expression. The key is to use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).

Step 2: Key Formula or Approach:
1. Start with the given equation: \(\cos \theta + \cos^2 \theta = 1\).
2. Rearrange it to express one function in terms of another.
3. Substitute this relationship into the expression we need to evaluate: \(\sin^2 \theta + \sin^4 \theta\).

Step 3: Detailed Explanation:
From the given equation:
\[ \cos \theta + \cos^2 \theta = 1 \] Rearrange it by subtracting \(\cos^2 \theta\) from both sides:
\[ \cos \theta = 1 - \cos^2 \theta \] Using the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute \(\sin^2 \theta\) into the equation:
\[ \cos \theta = \sin^2 \theta (*) \] Now, let's look at the expression we need to find:
\[ \sin^2 \theta + \sin^4 \theta \] This can be written as:
\[ \sin^2 \theta + (\sin^2 \theta)^2 \] From our derived relationship (*), we know that \(\sin^2 \theta = \cos \theta\). Let's substitute this into the expression:
\[ = (\cos \theta) + (\cos \theta)^2 \] \[ = \cos \theta + \cos^2 \theta \] From the original given equation, we know that \(\cos \theta + \cos^2 \theta = 1\).
Therefore, \(\sin^2 \theta + \sin^4 \theta = 1\).

Step 4: Final Answer:
The value of \(\sin^2 \theta + \sin^4 \theta\) is 1.