Question

use double and half angle formulas to find the equivalent expression to the following: (\frac{2sin x}{sin 2x}) select one:
○ a. (csc x)
○ b. (1)
○ c. (2cos x)
○ d. (sec x)
find the coordinates for ((x,y)) for the following set of system of equations.
(-1 - x = -3y)
(21x + 21 = 21y)
select one:
○ a. ((3,0))
○ b. ((0,-1))
○ c. ((0,3))
○ d. ((-1,0))

Explanation:

First Question (Trigonometric Expression)

Step1: Recall Double - Angle Formula

The double - angle formula for sine is $\sin2x = 2\sin x\cos x$.

Step2: Substitute into the Expression

We have the expression $\frac{2\sin x}{\sin2x}$. Substitute $\sin2x = 2\sin x\cos x$ into the expression:
$\frac{2\sin x}{2\sin x\cos x}$

Step3: Simplify the Expression

Cancel out the common factors $2\sin x$ (assuming $\sin x
eq0$ and $\cos x
eq0$ for the domain of the original function and the simplified function). We get $\frac{1}{\cos x}$, and $\frac{1}{\cos x}=\sec x$.

Second Question (System of Equations)

Step1: Simplify the Second Equation

Given the second equation $21x + 21=21y$, divide both sides of the equation by $21$:
$x + 1=y$, which can be rewritten as $y=x + 1$.

Step2: Substitute into the First Equation

The first equation is $-1-x=-3y$. Substitute $y = x + 1$ into the first equation:
$-1-x=-3(x + 1)$

Step3: Expand and Solve for $x$

Expand the right - hand side: $-1-x=-3x-3$
Add $3x$ to both sides: $-1 + 2x=-3$
Add $1$ to both sides: $2x=-2$
Divide both sides by $2$: $x=- 1$

Step4: Find the Value of $y$

Substitute $x=-1$ into $y=x + 1$: $y=-1 + 1=0$? Wait, no, wait. Wait, let's re - check. Wait, the first equation: $-1-x=-3y$, the second equation after simplification: $x + 1=y$ or $y=x + 1$. Wait, maybe I made a mistake in substitution. Let's start over.

First equation: $-1-x=-3y$ can be rewritten as $x + 1 = 3y$ (multiply both sides by - 1). Second equation: $21x+21 = 21y$ simplifies to $x + 1=y$. Now we have the system:

$$\begin{cases}x + 1=3y\\x + 1=y\end{cases}$$

Set the two expressions for $x + 1$ equal to each other: $3y=y$
Subtract $y$ from both sides: $2y = 0\Rightarrow y = 0$? No, that's not right. Wait, original first equation: $-1-x=-3y\Rightarrow x+1 = 3y$. Second equation: $21x + 21=21y\Rightarrow x + 1=y$. So substitute $y=x + 1$ into $x + 1=3y$: $x + 1=3(x + 1)$
$x + 1=3x+3$
$x-3x=3 - 1$
$-2x=2$
$x=-1$
Then $y=x + 1=-1 + 1=0$? But that's not one of the options. Wait, maybe I misread the first equation. The first equation is $-1 - x=-3y$, let's rewrite it as $x+1 = 3y$. The second equation: $21x + 21=21y\Rightarrow x + 1=y$. So we have $3y=y\Rightarrow 2y = 0\Rightarrow y = 0$, then $x=-1$. But the option d is $(-1,0)$. Wait, maybe I made a mistake in the first reading. Let's check the options again. Option d is $(-1,0)$. Let's verify:

For $x=-1,y = 0$:

First equation: $-1-(-1)=-3(0)\Rightarrow-1 + 1=0\Rightarrow0 = 0$, which is true.

Second equation: $21(-1)+21=21(0)\Rightarrow-21 + 21=0\Rightarrow0 = 0$, which is true.

Wait, earlier I thought option b was $(0,-1)$, but that was a mistake. Let's re - do the second question:

Step1: Simplify the equations

Equation 1: $-1-x=-3y\Rightarrow x + 1=3y$

Equation 2: $21x + 21=21y\Rightarrow x + 1=y$

Step2: Substitute $y=x + 1$ into $x + 1=3y$

$x + 1=3(x + 1)$

$x + 1=3x+3$

$x-3x=3 - 1$

$-2x=2$

$x=-1$

Step3: Find $y$

Substitute $x = - 1$ into $y=x + 1$, we get $y=-1 + 1=0$

So the solution is $(-1,0)$, which is option d.

Wait, there was a mistake in the initial substitution. So the correct answer for the second question is d. $(-1,0)$

Answer:

d. $\sec x$