Trigonometric Question sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x)

In summary, the conversation discussed simplifying the expression sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x) using the definitions of cosec(x) and sec(x). The final simplified expression is -sec(x).
  • #1
Yazan975
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In this question, I tried this:

sin^2(180-x) cosec(270+x) + cos^2(360-x) sec(180-x), where cosec(x) = 1/sin(x) and sec(x) = 1/cos(x)

-sin^2(180-x) = sin^2(x) and cos^2(x) = cos^2(x)

-The sin^2 and the 1/sin(x) cancle out along with the cos^2 and the 1/cos(x)

Therefore, I am left with sin(x)(270+x) + cos(x)(180-x)

This looks wrong. The answer on the answer sheet is -sec(x). I ask you for help please.
 

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  • #2
$\csc(270+x) = \dfrac{1}{\sin(270+x)} = \dfrac{1}{\sin(270)\cos{x}+\cos(270)\sin{x}} = \dfrac{1}{-\cos{x}}$

$\sec(180-x) = \dfrac{1}{\cos(180-x)} = \dfrac{1}{\cos(180)\cos{x} + \sin(180)\sin{x}} = \dfrac{1}{-\cos{x}}$

...

$\dfrac{\sin^2{x}}{-\cos{x}} + \dfrac{\cos^2{x}}{-\cos{x}}$

can you finish from here?
 

1. What is the value of sin^2(180-x)?

The value of sin^2(180-x) is equal to 1.

2. What is the value of cosec(270+x)?

The value of cosec(270+x) is undefined, as the cosecant function is undefined at 270+x.

3. What is the value of cos^2(360-x)?

The value of cos^2(360-x) is equal to 1.

4. What is the value of sec(180-x)?

The value of sec(180-x) is equal to -1.

5. What is the overall value of the trigonometric expression?

The overall value of the trigonometric expression is 1 + (-1) + 1 + (-1) = 0.

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