Final answer:
The equation cosec(x) - sin(x) = cos(x) cot(x) is shown to be true by manipulating the terms using trigonometric identities. However, the equation cannot be solved for x=0 as cot(0) is undefined.
Step-by-step explanation:
The student is asking to show that cosec(x) - sin(x) = cos(x) cot(x) and solve for a specific case when x=0, where x cannot be 180 degrees. We can start by expressing cosec(x) in terms of sin(x) and then try to transform the left-hand side of the equation into the right-hand side. cosec(x) = 1/sin(x)
Using this definition, we rearrange the left-hand side of the original equation: cosec(x) - sin(x) = 1/sin(x) - sin(x)
To combine these terms under a common denominator, we get:(1 - sin^2(x)) / sin(x)
Now, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite 1 - sin^2(x) as cos^2(x): cos^2(x) / sin(x)
Next, to express this in terms of cos(x) cot(x), we use the identity that cot(x) = cos(x)/sin(x):
cos^2(x) / sin(x) = cos(x) (cos(x)/sin(x))
This simplifies to:
cos(x) cot(x)
Thus, we have shown that cosec(x) - sin(x) equals to cos(x) cot(x).
When solving for x=0, we must remember that cot(0) is undefined since it involves division by zero (sin(0)=0). Therefore, the equation cannot be solved for x=0 as it is not within the domain where the trigonometric functions involved are defined.