Prove that: (sin x - 2 sin³ x)/(2 cos³ x - cos x) = tan x​

[tex]\large\underline{\sf{Solution-}}[/tex]

We have to prove that:

[tex]\rm\longmapsto \dfrac{ \sin(x) - 2 \sin^{3} (x) }{2 \cos^{3} (x) - \cos(x) } = \tan(x) [/tex]

Taking LHS, we get:

[tex]\rm = \dfrac{ \sin(x) - 2 \sin^{3} (x) }{2 \cos^{3} (x) - \cos(x) }[/tex]

We can take sin(x) as common from numerator part. We get:

[tex]\rm = \dfrac{ \sin(x) \{1- 2 \sin^{2} (x) \}}{2 \cos^{3} (x) - \cos(x) }[/tex]

Similarly, we can take cos(x) as common from first two terms. We get:

[tex]\rm = \dfrac{ \sin(x) \{1- 2 \sin^{2} (x) \}}{ \cos(x) \{ 2 \cos^{2} (x) -1\}}[/tex]

[tex]\rm = \dfrac{ \sin(x) }{ \cos(x) } \cdot \dfrac{\{1- 2 \sin^{2} (x) \}}{\{ 2 \cos^{2} (x) -1\}}[/tex]

As we know that:

[tex]\rm\longmapsto \dfrac{ \sin(x) }{ \cos(x) } = \tan(x) [/tex]

We get:

[tex]\rm = \tan(x) \cdot \dfrac{\{1- 2 \sin^{2} (x) \}}{\{ 2 \cos^{2} (x) -1\}}[/tex]

Now, we will expand the terms in fraction:

[tex]\rm = \tan(x) \cdot \dfrac{\{1- \sin^{2} (x) - { \sin }^{2}(x) \}}{\{\cos^{2} (x) + \cos^{x}(x) -1\}}[/tex]

As we know that:

[tex]\rm \longmapsto \sin^{2}(x)+cos^{2}(x)=1[/tex]

[tex]\rm \longmapsto\cos^{2}(x)=1-sin^{2}(x)[/tex]

[tex]\rm \longmapsto\cos^{2}(x) - 1=-sin^{2}(x)[/tex]

Now substitute the values in the expression, we get:

[tex]\rm = \tan(x)\cdot \dfrac{ \cos^{2} (x)-{ \sin }^{2}(x)}{\cos^{2} (x) - \sin^{2} (x) }[/tex]

[tex]\rm = \tan(x)\cdot1[/tex]

[tex]\rm = \tan(x)[/tex]

Here, we notice that LHS = RHS

Proved.

(02.05 MC)
Graph g(x), where f(x) = 4x - 2 and g(x) = f(x + 1).

What does the initial amount represent on the linear equations word problems?

what is the state income tax deducted from gross pay of $864 with state income tax of 4%?​

6846 is shared equally between 7 people how much does each person get?​

please help me with this sum​

A water tank is a cube of side 2m the depth of water in it is60cm. What is the volume of water in the tank?

3/5 of 2 kg
what is the answer​

6 = 2n + 10
Please help

4✓6 × 3✓6
multiply and simplify
show work​

The line, L1 passes through the points A (-3,-2) and B (3,-8) Find the gradient of L1.​

Evaluate the solid bounded by 2x + z = 2 and (x − 1)^2 + y^2 = z. Sketch the given solid.

A caterer charges 5500 plus 530 per guest to cater a wedding. Walt and Traci don't want to spend more than $8000 on catering Write
and solve an inequality in terms of the number ofguests, 9 that can be invited.

HELPPPPPPPpppppppppppppppppp

The Foldgers Black Silk K-cup coffee pods weigh on average .28 ounces with a standard deviation of .05 ounces. Answer each of the questions.
68%, 2.5%, 81.5%, 95%, 41-42, 20-21, 16%, 84%, 121-122, 5%, 87-88, 64, 35-36
What percentage of K-cups can be expected to fall within 2 Standard Deviations of the mean?

What is the probability that a K-cup will have more than .33 ounces?

Sam's club sells K-cups in a box of 128. How many of them will weigh less than .23 ounces?

K-cups that are more than 2 SD below the mean are considered to be to "weak". What percentage of K-cup are considered to "weak"

In a standard box of 44 k-cups, how many will weigh between .23 and .38 ounces

The ideal weight for K-cups are within 1 SD of the mean, how many of the Sam's Club box of K-cups are ideal?

Tallahassee, FL has a population of 483,493. New York, NY has 345,876 more people than Tallahassee. Phoenix, AZ has 279,426 less people than New York.

b.) Round to the nearest hundred thousand to check the reasonableness of your answer for the population of Phoenix, AZ.

3.1.PS-16
Question Help
o
A new health drink has 170% of the recommended daily allowance (RDA) for a certain vitamin. The
RDA for this vitamin is 30 mg. How many milligrams of the vitamin are in the drink?
There are
mg of the vitamin in the drink.

Help I have a C in math. I don't understand this Alegbra. If you get it right then I will give you Brainlist! (Ignore the black dots because I am not sure if they are right)​

Find the percent of change from

10 yards to 15 yards
1. 5%
2. 33%
3. 15%
4. 50%

Please help me asappp

Help me please!!!!! ✨✨✨✨✨✨✨✨

What is the product of 2.5 x 4.3?

Find the equation of the linear function represented by the table

y 6 9 12 15
x 1 2 3 4






​

Using the Addition method, solve for y in the following system of linear equations .
3x - y=1
X + 2y=5

Karan uses
of a tin of baby food each day for her baby.
Karan and her baby are going on a 7-day holiday.
What is the least number of tins of baby food that Karan must bring with her?
Show your working.

create a equation for this exponential function

A car travelling at 120km per hour. How many metres does the car travel in 2 1/2 minutes

pls help I will put brainliest​

Sarah is buying some school clothes.
She found a sweater she likes that costs $18.85 and some t-shirts that are on sale for $4 each.
She got several t shirts of different colors and spent a total of $54.85.
Which equation describes the situation where x is the number of t shirts?

What is the slope between these two points (-2, 7) and (4, 6)?(

According to the following image, what is the length of ST?