Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To solve the equation [tex]\( \frac{\cot^2 t}{\csc t} = \csc t - \sin t \)[/tex], let's break it down step by step, considering trigonometric identities and simplifications.
First, recall the definitions of the trigonometric functions involved:
[tex]\[ \cot t = \frac{\cos t}{\sin t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Given the left-hand side (LHS) of the equation:
[tex]\[ \frac{\cot^2 t}{\csc t} \][/tex]
Substitute the definitions of [tex]\(\cot t\)[/tex] and [tex]\(\csc t\)[/tex]:
[tex]\[ \cot^2 t = \left( \frac{\cos t}{\sin t} \right)^2 = \frac{\cos^2 t}{\sin^2 t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Now substitute these into the left-hand side:
[tex]\[ \frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \cdot \sin t = \frac{\cos^2 t \cdot \sin t}{\sin^2 t} = \frac{\cos^2 t}{\sin t} \][/tex]
So the simplified form of the left-hand side is:
[tex]\[ \frac{\cos^2 t}{\sin t} \][/tex]
Now consider the right-hand side (RHS) of the equation:
[tex]\[ \csc t - \sin t \][/tex]
Using the definition of [tex]\(\csc t\)[/tex]:
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Thus, the right-hand side can be expressed as:
[tex]\[ \frac{1}{\sin t} - \sin t \][/tex]
So, we have the simplified forms:
[tex]\[ \text{LHS} = \frac{\cos^2 t}{\sin t} \][/tex]
[tex]\[ \text{RHS} = \frac{1}{\sin t} - \sin t \][/tex]
Now we can write the given equation in its simplified form:
[tex]\[ \frac{\cos^2 t}{\sin t} = \frac{1}{\sin t} - \sin t \][/tex]
Therefore, the simplified forms of both sides of the equation are:
[tex]\[ \boxed{\left( \frac{\cos^2 t}{\sin t}, \frac{1}{\sin t} - \sin t \right)} \][/tex]
First, recall the definitions of the trigonometric functions involved:
[tex]\[ \cot t = \frac{\cos t}{\sin t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Given the left-hand side (LHS) of the equation:
[tex]\[ \frac{\cot^2 t}{\csc t} \][/tex]
Substitute the definitions of [tex]\(\cot t\)[/tex] and [tex]\(\csc t\)[/tex]:
[tex]\[ \cot^2 t = \left( \frac{\cos t}{\sin t} \right)^2 = \frac{\cos^2 t}{\sin^2 t} \][/tex]
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Now substitute these into the left-hand side:
[tex]\[ \frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \cdot \sin t = \frac{\cos^2 t \cdot \sin t}{\sin^2 t} = \frac{\cos^2 t}{\sin t} \][/tex]
So the simplified form of the left-hand side is:
[tex]\[ \frac{\cos^2 t}{\sin t} \][/tex]
Now consider the right-hand side (RHS) of the equation:
[tex]\[ \csc t - \sin t \][/tex]
Using the definition of [tex]\(\csc t\)[/tex]:
[tex]\[ \csc t = \frac{1}{\sin t} \][/tex]
Thus, the right-hand side can be expressed as:
[tex]\[ \frac{1}{\sin t} - \sin t \][/tex]
So, we have the simplified forms:
[tex]\[ \text{LHS} = \frac{\cos^2 t}{\sin t} \][/tex]
[tex]\[ \text{RHS} = \frac{1}{\sin t} - \sin t \][/tex]
Now we can write the given equation in its simplified form:
[tex]\[ \frac{\cos^2 t}{\sin t} = \frac{1}{\sin t} - \sin t \][/tex]
Therefore, the simplified forms of both sides of the equation are:
[tex]\[ \boxed{\left( \frac{\cos^2 t}{\sin t}, \frac{1}{\sin t} - \sin t \right)} \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.