Prove that the given equation is not an identity by finding a value of x for which the two sides have different values.

cosine xcosxequals=1 minus sine x1−sinx

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Part 1

Choose the correct value of x for which the two sides have different values. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

A.

The equation is not an identity. When

xequals=00​,

cosine xcosxequals=enter your response here

and

1 minus sine x1−sinxequals=enter your response here.

B.

The equation is not an identity. When

xequals=piπ​,

cosine xcosxequals=enter your response here

and

1 minus sine x1−sinxequals=enter your response here.

C.

The equation is not an identity. When

xequals=StartFraction pi Over 2 EndFractionπ2​,

cosine xcosxequals=enter your response here

and

1 minus sine x1−sinxequals=enter your response here.

We are given the equation:

\[
\cos(x) = 1 - \sin(x)
\]

Our task is to find a value of \(x\) for which the two sides of the equation are different, proving that it is not an identity.

Let’s evaluate the left side (\(\cos(x)\)) and the right side (\(1 - \sin(x)\)) for the given options and compare their values.

### Option A: \(x = 0\)
1. Left-hand side: \(\cos(0) = 1\)
2. Right-hand side: \(1 - \sin(0) = 1 - 0 = 1\)

For \(x = 0\), both sides are equal. So this option does not show a difference between the two sides.

### Option B: \(x = \pi\)
1. Left-hand side: \(\cos(\pi) = -1\)
2. Right-hand side: \(1 - \sin(\pi) = 1 - 0 = 1\)

For \(x = \pi\), the left side is \(-1\) and the right side is \(1\). Since \(-1 \neq 1\), this option shows that the equation is not an identity.

### Option C: \(x = \frac{\pi}{2}\)
1. Left-hand side: \(\cos\left(\frac{\pi}{2}\right) = 0\)
2. Right-hand side: \(1 - \sin\left(\frac{\pi}{2}\right) = 1 - 1 = 0\)

For \(x = \frac{\pi}{2}\), both sides are equal to 0. So this option also does not show a difference between the two sides.

### Conclusion:
The correct choice is **Option B**. When \(x = \pi\), \(\cos(x) = -1\) and \(1 - \sin(x) = 1\), so the equation is not an identity.

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Would you like more details or have any questions? 

Here are 5 related questions for further exploration:
1. What is the general definition of a trigonometric identity?
2. Can you solve for \(x\) in the equation \(\cos(x) = 1 - \sin(x)\)?
3. How can we graphically verify that \(\cos(x)\) and \(1 - \sin(x)\) are not identical functions?
4. How do we find the zeros of trigonometric functions like \(\cos(x)\) and \(\sin(x)\)?
5. How does the value of \(\sin(x)\) relate to \(\cos(x)\) through the Pythagorean identity?

**Tip:** When proving that an equation is not an identity, substituting specific values of \(x\) can quickly show whether the two sides are always equal or not.

Prove that the given equation is not an identity by finding a value of x for which the two sides have different values.

cos(x) = 1 - sin(x)

Choose the correct value of x for which the two sides have different values. Select the correct choice below:
A. The equation is not an identity when x = 0.
B. The equation is not an identity when x = π.
C. The equation is not an identity when x = π/2.

Explore how to prove that the trigonometric equation cos(x) = 1 - sin(x) is not an identity by finding the value of x where both sides differ. This step-by-step solution demonstrates key trigonometric concepts and is suitable for students in Grades 9-12.

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