sin t /1-cos t - 1+ cos t/sin t= 0
sin t /(1-cos t) - (1+cos t)/sin t
sin t (1+cos t)/((1-cos t)(1+cos t)) - (1+cos t)/sin t
sin t (1+cos t)/(1-cos^2 t) - (1+cos t)/sin t
sin t (1+cos t)/sin^2 t - sin t (1+cos t)/sin^2 t
0
To solve the equation (sin t / (1 - cos t)) - ((1 + cos t) / sin t) = 0, we can follow these steps:
Step 1: Simplify the expression
Let's start by finding a common denominator for the two fractions. The denominator can be sin t * (1 - cos t). Now, we can rewrite the equation:
(sin t * sin t - (1 + cos t) * (1 - cos t)) / (sin t * (1 - cos t)) = 0
Expanding the numerator:
(sin^2 t - (1 - cos^2 t)) / (sin t * (1 - cos t)) = 0
Using the identity sin^2 t + cos^2 t = 1:
((1 - cos^2 t) - (1 - cos^2 t)) / (sin t * (1 - cos t)) = 0
Simplifying:
0 / (sin t * (1 - cos t)) = 0
Since the numerator is 0, the equation simplifies to:
0 = 0
Step 2: Analyze the result
The equation 0 = 0 is true for any value of t. This means that any value of t will satisfy the original equation, making the equation an identity.
Therefore, the solution to the equation sin t / (1 - cos t) - (1 + cos t) / sin t = 0 is t belongs to the set of all real numbers.
In summary, the equation is an identity, and any value of t will satisfy it.