Respuesta :
Using trigonometric identities, the simplified expression, without quotients, is given as follows:
A. [tex]\cos{\beta}\cot{\beta}[/tex]
Which trigonometric identities are used to solve this question?
The cosecant of an angle is given by one divided by the sine of the angle, hence:
[tex]\csc{\beta} = \frac{1}{\sin{\beta}}[/tex]
The sine and the cosine of an angle are related as follows:
[tex]\sin^2{\beta} + \cos^2{\beta} = 1[/tex]
[tex]\cos^2{\beta} = 1 - \sin^2{\beta}[/tex]
The cotangent of an angle is:
[tex]\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}[/tex]
Which is the simplified expression?
The expression given is:
[tex]\csc{\beta} - \sin{\beta}[/tex]
Simplifying the cosecant:
[tex]\csc{\beta} - \sin{\beta} = \frac{1}{\sin{\beta}} - \sin{\beta} = \frac{1 - \sin^{2}\beta}{\sin{beta}}[/tex]
Then, applying the last two identities in the subsection above to remove the quotient, the simplified expression will be found as follows:
[tex]\frac{1 - \sin^{2}\beta}{\sin{beta}} = \frac{\cos^2{\beta}}{\sin{\beta}} = \frac{\cos{\beta}}{\sin{\beta}} \times \cos{\beta} = \cos{\beta}\cot{\beta}[/tex]
Hence option A is correct.
More can be learned about trigonometric identities at https://brainly.com/question/26676095
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