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3 years ago

Verify the following trigonometric identity: (csc x + cot x)^2 = cos x +1/ 1- cos x

Mathematics

1 answer:

Airida [17]3 years ago

3 0

9514 1404 393

Explanation:

As written, the equation is not an identity. Perhaps you want to show ...

(csc(x) +cot(x))² = (cos(x) +1)/(1 -cos(x))

We will transform the left-side expression to the form of the right-side expression.

$(\csc(x)+\cot(x))^2=\left(\dfrac{1}{\sin(x)}+\dfrac{\cos(x)}{\sin(x)}\right)^2=\dfrac{(1+\cos(x))^2}{\sin(x)^2}\\\\=\dfrac{(1+\cos(x))^2}{1-\cos(x)^2}=\dfrac{1+\cos(x)}{1-\cos(x)}\cdot\dfrac{1+\cos(x)}{1+\cos(x)}=\boxed{\dfrac{\cos(x)+1}{1-\cos(x)}}$

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Answer:

$15.2\ units$

Step-by-step explanation:

step 1

Plot the vertices of the polygon to better understand the problem

we have

$A(-2, 1). B(-3, 3), C(-1, 5), D(2, 4),E(2, 1)$

using a graphing tool

The polygon is a pentagon (the number of sides is 5)

see the attached figure

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Find the distance AB

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$d_A_B=\sqrt{5}=2.24\ units$

step 3

Find the distance BC

$B(-3, 3), C(-1, 5)$

substitute in the formula

$d=\sqrt{(5-3)^{2}+(-1+3)^{2}}$

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step 4

Find the distance CD

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substitute in the formula

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$d=\sqrt{(-1)^{2}+(3)^{2}}$

$d_C_D=\sqrt{10}=3.16\ units$

step 5

Find the distance DE

$D(2, 4),E(2, 1)$

substitute in the formula

$d=\sqrt{(1-4)^{2}+(2-2)^{2}}$

$d=\sqrt{(-3)^{2}+(0)^{2}}$

$d_D_E=\sqrt{9}\ units$

$d_D_E=3\ units$

step 6

Find the distance AE

$A(-2, 1).E(2, 1)$

substitute in the formula

$d=\sqrt{(1-1)^{2}+(2+2)^{2}}$

$d=\sqrt{(0)^{2}+(4)^{2}}$

$d_A_E=\sqrt{16}\ units$

$d_A_E=4\ units$

step 7

Find the perimeter

$P=AB+BC+CD+DE+AE$

substitute the values

$P=2.24+2.83+3.16+3+4=15.23\ units$

Round to the nearest tenth of a unit

$P=15.2\ units$

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