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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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gavmur [86]
Hello!
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Exercise questions
balu736 [363]

Answer:

2.

a)1

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Step-by-step explanation:

There's some identity trigonometric equation, which are valid for all angles,and they doesn't depends on the measure of angle!

some of em are follows:. (x is the given angle)

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You can remember these identity, its gonna help alot.

now back to question,. {x is representing angles)

for (a) sin(x)^2+cos(x)^2=1, this is true for all x, dat means that for all the angle given in question(for ,15°,30°,45°,60° and 120°),we will get 1

for(b) ,

cosec(x)^2=1+cot(x)^2

i.e, cosec(x)^2-cot(x)^2=1, again this is true for all x dat means that for all the angle given in question ,we will get 1

for (c),

sec(x)^2=1+tan(x)^2

i.e,sec(x)^2-tan(x)^2=1,again this is true for all x, dat means that for all the angle given in question ,we will get 1

✌️:)

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faust18 [17]

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Step-by-step explanation:

40+x = 4x-3

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43+x = 4x

<u>-</u><u>x</u> <u>-</u><u>x</u>

<u>43</u> = <u>3x</u>

3 3

14.3 = x

then substitude this on 5x+3 on <em>W</em><em>T</em> and you arr good

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const2013 [10]

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Step-by-step explanation:

1. Write the equation 2x-5+4x-1+30=180 because all triangle angles add up to 180°

2. Simplify the equation by combining like terms. Write the equation as 6x+24=180

3. Subtract 24 from both sides. Write the equation as 6x=156.

4. Divide 156/6 and get 26. That is your answer (x=26).

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