Question

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

• A. 2
• B. 1
• C. -2
• D. 0

Hint:

1. Given: \(\sin A + \sin^2 A = 1\). 2. Rearrange: \(\sin A = 1 - \sin^2 A\). 3. Use identity: \(1 - \sin^2 A = \cos^2 A\). 4. So, from (2) and (3): \(\sin A = \cos^2 A\). 5. We need to find \(\cos^2 A + \cos^4 A\). This can be written as \(\cos^2 A + (\cos^2 A)^2\). 6. Substitute \(\cos^2 A = \sin A\): Expression becomes \(\sin A + (\sin A)^2 = \sin A + \sin^2 A\). 7. From the given condition (1), \(\sin A + \sin^2 A = 1\). Thus, \(\cos^2 A + \cos^4 A = 1\).

Answer 1

The Correct Option is B

Concept: This problem uses the fundamental trigonometric identity \(\sin^2 A + \cos^2 A = 1\). Step 1: Manipulate the given equation We are given: \( \sin A + \sin^2 A = 1 \). Rearrange this equation to isolate \(\sin A\): \[ \sin A = 1 - \sin^2 A \] Step 2: Use the fundamental identity We know that \(\sin^2 A + \cos^2 A = 1\). From this identity, we can write \(\cos^2 A = 1 - \sin^2 A\). Step 3: Substitute to find a relationship between \(\sin A\) and \(\cos^2 A\) From Step 1, we have \( \sin A = 1 - \sin^2 A \). From Step 2, we have \( \cos^2 A = 1 - \sin^2 A \). Since both \(\sin A\) and \(\cos^2 A\) are equal to \(1 - \sin^2 A\), we can conclude: \[ \sin A = \cos^2 A \] Step 4: Evaluate the expression \( \cos^2 A + \cos^4 A \) We need to find the value of \( \cos^2 A + \cos^4 A \). We can write \(\cos^4 A\) as \((\cos^2 A)^2\). So the expression is \( \cos^2 A + (\cos^2 A)^2 \). From Step 3, we found that \( \cos^2 A = \sin A \). Substitute this into the expression: \[ \cos^2 A + \cos^4 A = \sin A + (\sin A)^2 \] \[ \cos^2 A + \cos^4 A = \sin A + \sin^2 A \] Step 5: Use the original given condition The original given condition was \( \sin A + \sin^2 A = 1 \). Therefore, \[ \cos^2 A + \cos^4 A = 1 \] The value of the expression is 1.