Is it possible to write the question of every line in slope intercept form

Hello! Verify the identity. Please show your work! Use trigonometric identities to verify each expression is equal.

RSB [31]

Answer:

See Below.

Step-by-step explanation:

We want to verify the identity:

$\displaystyle \csc^2 x -2\csc x \cot x +\cot ^2 x = \tan^2\left(\frac{x}{2}\right)$

Note that the left-hand side is a perfect square trinomial pattern. Namely:

$a^2-2ab+b^2=(a-b)^2$

If we let a = csc(x) and b = cot(x), we can factor it as such:

$\displaystyle (\csc x - \cot x)^2 = \tan^2\left(\frac{x}{2}\right)$

Let csc(x) = 1 / sin(x) and cot(x) = cos(x) / sin(x):

$\displaystyle \left(\frac{1}{\sin x}-\frac{\cos x }{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)$

Combine fractions:

$\displaystyle \left(\frac{1-\cos x}{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)$

Square (but do not simplify yet):

$\displaystyle \frac{(1-\cos x)^2}{\sin ^2x}=\tan^2\left(\frac{x}{2}\right)$

Now, we can make a substitution. Let u = x / 2. So, x = 2u. Substitute:

$\displaystyle \frac{(1-\cos 2u)^2}{\sin ^22u}=\tan^2u$

Recall that cos(2u) = 1 - sin²(u). Hence:

$\displaystyle \frac{(1-(1-2\sin^2u))^2}{\sin ^2 2u}=\tan^2u$

Simplify:

$\displaystyle \frac{4\sin^4 u}{\sin ^2 2u}=\tan^2 u$

Recall that sin(2u) = 2sin(u)cos(u). Hence:

$\displaystyle \frac{4\sin^4 u}{(2\sin u\cos u)^2}=\tan^2 u$

Square:

$\displaystyle \frac{4\sin^4 u}{4\sin^2 u\cos ^2u}=\tan^2 u$

Cancel:

$\displaystyle \frac{\sin ^2 u}{\cos ^2 u}=\tan ^2 u$

Since sin(u) / cos(u) = tan(u):

$\displaystyle \left(\frac{\sin u}{\cos u}\right)^2=\tan^2u=\tan^2u$

We can substitute u back for x / 2:

$\displaystyle \tan^2\left(\frac{x}{2}\right)= \tan^2\left(\frac{x}{2}\right)$

Hence proven.

3 0

3 years ago

2,000 water bottles are being labeled and boxed. Each box holds 6 water bottles. How many bottles will be left over?

Lostsunrise [7]

Answer:

2 left out, with 333 boxes

Step-by-step explanation:

2000/6=333.333333 ( $\frac{1}{3}$ )

$\frac{1}{3}$ of 6 is 2. Therefore, there would be 333 boxes, with two left unboxed.

4 0

3 years ago

Read 2 more answers

A hotel Manager needed to seat 250 guests at a wedding party. Can he seat them in groups of 5 or 10?

svet-max [94.6K]

The manager can do both but if he does 10 he will have 25 groups but if he does 5 he will have 50 groups so the simple way is 10 so he will have 25 groups

8 0

4 years ago

It took Azul 1.75 hours to bike The Blue Mountain Trail. Azul averaged 12.5 miles per hour. What is the best estimate for the le

Kryger [21]

The answer is 26 because if you round 12.5 it would be 13 and of you would round 1.75 it would be 2. After you do this you would multiply the 2 numbers and get 26 which is your answer.

5 0

3 years ago

Read 2 more answers

The point-slope form of a line that has a slope of Linear Equation and passes through the point (3, 0) is shown.

seraphim [82]

Answer:

y=1/4x-3/4

Step-by-step explanation:

hope this helped

6 0

4 years ago

Read 2 more answers