Questions: Move the points to change the order of the lines in the proof to show that (1-cos^2(θ))(1+cot^2(θ))=1 is true starting with something known and so each line of the proof follows from the previous line. sin^2(θ)(1+cot^2(θ))=1 (sin^2(θ)+cos^2(θ)-cos^2(θ))(1+cot^2(θ))=1 sin^2(θ)+cos^2(θ)=1 sin^2(θ)(1+(cos^2(θ)/sin^2(θ)))=1 (1-cos^2(θ))(1+cot^2(θ))=1

Move the points to change the order of the lines in the proof to show that (1-cos^2(θ))(1+cot^2(θ))=1 is true starting with something known and so each line of the proof follows from the previous line.
sin^2(θ)(1+cot^2(θ))=1
(sin^2(θ)+cos^2(θ)-cos^2(θ))(1+cot^2(θ))=1
sin^2(θ)+cos^2(θ)=1
sin^2(θ)(1+(cos^2(θ)/sin^2(θ)))=1
(1-cos^2(θ))(1+cot^2(θ))=1

Transcript text: Move the points to change the order of the lines in the proof to show that $\left(1-\cos ^{2}(\theta)\right)\left(1+\cot ^{2}(\theta)\right)=1$ is true starting with something known and so each line of the proof follows from the previous line. $\sin ^{2}(\theta)\left(1+\cot ^{2}(\theta)\right)=1$ $\left(\sin ^{2}(\theta)+\cos ^{2}(\theta)-\cos ^{2}(\theta)\right)\left(1+\cot ^{2}(\theta)\right)=1$ $\sin ^{2}(\theta)+\cos ^{2}(\theta)=1$ $\sin ^{2}(\theta)\left(1+\frac{\cos ^{2}(\theta)}{\sin ^{2}(\theta)}\right)=1$ $\left(1-\cos ^{2}(\theta)\right)\left(1+\cot ^{2}(\theta)\right)=1$

Solution

Solution Steps

To show that $(1 - \cos^2(\theta))(1 + \cot^2(\theta)) = 1$, we need to start with known trigonometric identities and manipulate them step by step to reach the desired equation. Here is the correct order of the lines in the proof:

$\sin^2(\theta) + \cos^2(\theta) = 1$ (Pythagorean identity)
$\sin^2(\theta) = 1 - \cos^2(\theta)$ (Rearranging the Pythagorean identity)
$\sin^2(\theta)(1 + \cot^2(\theta)) = 1$ (Starting with the left-hand side of the desired equation)
$\sin^2(\theta)\left(1 + \frac{\cos^2(\theta)}{\sin^2(\theta)}\right) = 1$ (Expressing $\cot^2(\theta)$ as $\frac{\cos^2(\theta)}{\sin^2(\theta)}$)
$\left(\sin^2(\theta) + \cos^2(\theta) - \cos^2(\theta)\right)\left(1 + \cot^2(\theta)\right) = 1$ (Simplifying the expression)
$(1 - \cos^2(\theta))(1 + \cot^2(\theta)) = 1$ (Final step showing the desired equation)

Step 1: Pythagorean Identity

We start with the well-known Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This identity is fundamental in trigonometry and will be used to derive other expressions.

Step 2: Rearranging the Identity

From the Pythagorean identity, we can express $\sin^2(\theta)$ as: \[ \sin^2(\theta) = 1 - \cos^2(\theta) \]

Step 3: Left-Hand Side of the Desired Equation

We consider the left-hand side of the equation we want to prove: \[ \sin^2(\theta) \left(1 + \cot^2(\theta)\right) \] This expression can be rewritten using the definition of $\cot(\theta)$: \[ \cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)} \] Thus, we have: \[ \sin^2(\theta) \left(1 + \frac{\cos^2(\theta)}{\sin^2(\theta)}\right) \]

Step 4: Expanding the Expression

Expanding the expression gives: \[ \sin^2(\theta) + \cos^2(\theta) \] Using the Pythagorean identity, this simplifies to: \[ 1 \]

Step 5: Final Step

Now we can express the left-hand side of the original equation: \[ (1 - \cos^2(\theta))(1 + \cot^2(\theta)) \] Substituting $\cot^2(\theta)$ gives: \[ (1 - \cos^2(\theta))\left(\frac{\sin^2(\theta) + \cos^2(\theta)}{\sin^2(\theta)}\right) \] This simplifies to: \[ (1 - \cos^2(\theta)) \cdot \frac{1}{\sin^2(\theta)} = 1 \]