If `cos x+ cos ^(2) x=1,. ` then the value of ` sin ^(2) x+ sin ^(4) x `is

Options

A. 1
B. `-1`
C. 0
D. 2

Answer

To solve the problem, we start with the given equation:

**Step 1:** Given the equation:
\[ \cos x + \cos^2 x = 1 \]

**Step 2:** Rearranging the equation:
We can rearrange this equation to express \(\cos^2 x\):
\[ \cos^2 x = 1 - \cos x \]

**Step 3:** Substituting \(\sin^2 x\):
Using the Pythagorean identity, we know that:
\[ \sin^2 x = 1 - \cos^2 x \]
Substituting \(\cos^2 x\) from the previous step:
\[ \sin^2 x = 1 - (1 - \cos x) = \cos x \]

**Step 4:** Finding \(\sin^4 x\):
Now, we need to find \(\sin^4 x\):
\[ \sin^4 x = (\sin^2 x)^2 = (\cos x)^2 \]

**Step 5:** Now we can express \(\sin^2 x + \sin^4 x\):
\[ \sin^2 x + \sin^4 x = \cos x + \cos^2 x \]

**Step 6:** Substitute back the value from the original equation:
Since we know from the original equation that \(\cos x + \cos^2 x = 1\):
\[ \sin^2 x + \sin^4 x = 1 \]

Thus, the final answer is:
\[ \sin^2 x + \sin^4 x = 1 \]

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