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Rudik [331]
3 years ago
9

Prove the following

Mathematics
1 answer:
DerKrebs [107]3 years ago
7 0

\bf [cot(\theta )+csc(\theta )]^2=\cfrac{1+cos(\theta )}{1-cos(\theta )} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the left-hand side}}{[cot(\theta )+csc(\theta )]^2}\implies cot^2(\theta )+2cot(\theta )csc(\theta )+csc^2(\theta ) \\\\\\ \cfrac{cos^2(\theta )}{sin^2(\theta )}+2\cdot \cfrac{cos(\theta )}{sin(\theta )}\cdot \cfrac{1}{sin(\theta )}+\cfrac{1}{sin^2(\theta )}\implies \cfrac{cos^2(\theta )}{sin^2(\theta )}+\cfrac{2cos(\theta )}{sin^2(\theta )}+\cfrac{1}{sin^2(\theta )}

\bf \cfrac{\stackrel{\textit{perfect square trinomial}}{cos^2(\theta )+2cos(\theta )+1}}{sin^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{doing the right-hand-side}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}}\implies \stackrel{\textit{multiplying by the denominator's conjugate}}{\cfrac{1+cos(\theta )}{1-cos(\theta )}\cdot \cfrac{1+cos(\theta )}{1+cos(\theta )}}

\bf \cfrac{[1+cos(\theta )]^2}{\underset{\textit{difference of squares}}{[1-cos(\theta )][1+cos(\theta )]}}\implies \cfrac{[cos(\theta )+1]^2}{1^2-cos^2(\theta )} \\\\\\ \cfrac{[cos(\theta )+1]^2}{1-cos^2(\theta )}\implies \boxed{\cfrac{[cos(\theta )+1]^2}{sin^2(\theta )}}

recall that sin²(θ) + cos²(θ) = 1, thus sin²(θ) = 1 - cos²(θ).

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It takes me 12 minutes to run 2 kilometers. How long will it take me to run 10,000 meters?
galina1969 [7]
60min or 1 hour

Because 2km is equal to 2000m
Therefore you multiply 12 min by 5 (because 2000m x 5 is 10000m)

7 0
3 years ago
Can someone plz help me ?
nevsk [136]

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Step-by-step explanation:

4 0
2 years ago
A fruit company delivers its fruit in two types of boxes: large and small. A delivery of 5 large boxes and 3 small boxes has a t
-BARSIC- [3]

Answer:

The weight of large and small box are 18.25 kg and 13.25 kg respectively.

Step-by-step explanation:

Let the weight of large and small box are x and y respectively.

It is given that a delivery of 5 large boxes and 3 small boxes has a total weight of 131 kilograms.

5x+3y=131                .... (1)

A delivery of 2 large boxes and 6 small boxes has a total weight of 116 kilograms.

2x+6y=116                .... (2)

On solving (1) and (2) using graphing calculator, we get

x=18.25

y=13.25

Therefore the weight of large and small box are 18.25 kg and 13.25 kg respectively.

4 0
3 years ago
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If A= (6,11), B= (1,5), and C= (7,0), show by means of slopes that ABC is a right triangle. Name the hypotenuse.
nexus9112 [7]

Since the product of AB and BC is -1, hence the ABC is a right triangle and the length AC is the hypotenuse

<h3>Right angled triangle</h3>

A right angles triangle is a triangle that has 3 sides and angles with one of its angles to be 90 degrees.

For the given coordinates to be a right angled triangle, one of the line must be perpendicular to the other and for two lines to be perpendicular, the product of its slope must be -1.

Find the slopes of AB, BC and AC

Slope of AB = 5-11/1-6
Slope of AB = -6/-5 = 6/5

Slope of BC = 0-5/7-1
Slope of BC = -5/6

Slope of AC = 0-11/7-6
Slope of AC = -11/1. = -11

Take the product of AB and BC
AB * BC = 6/5 * -5/6
AB * BC = -1

Since the product of AB and BC is -1, hence the ABC is a right triangle and the length AC is the hypotenuse

Learn more on right angle triangle here: brainly.com/question/22790996

#SPJ1

6 0
2 years ago
Can someone plz help me with this one problem plz
ICE Princess25 [194]
8mm bc a=lw. If a=32 and l=4, you get 4w=32. Then, solve for w.
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