(1)
tan^2x(1+cot^2x)= _____________
(1-sin^2x)
To solve this equation, we can simplify each term step by step.
We'll start with the left-hand side of the equation: tan^2x(1+cot^2x).
First, let's recall the definitions of the tangent (tan) and cotangent (cot) functions:
- tan(x) = sin(x)/cos(x)
- cot(x) = cos(x)/sin(x)
Next, we can substitute these definitions into the equation:
tan^2x(1+cot^2x) = (sin(x)/cos(x))^2(1+(cos(x)/sin(x))^2)
Now, let's simplify this expression:
= sin^2(x)/cos^2(x) * (1 + cos^2(x)/sin^2(x))
To simplify further, we can use the identity: sin^2(x) + cos^2(x) = 1.
= (sin^2(x)/cos^2(x)) * (sin^2(x)/sin^2(x) + cos^2(x)/sin^2(x))
= (sin^2(x)/cos^2(x)) * (1/cos^2(x) + 1/sin^2(x))
Now, let's find a common denominator:
= (sin^2(x)/cos^2(x)) * (sin^2(x) + cos^2(x)) / (cos^2(x) * sin^2(x))
= (sin^2(x)/cos^2(x)) * (1) / (cos^2(x) * sin^2(x))
Using the reciprocal identity: a/b = 1/(b/a), we can rewrite this as:
= sin^2(x) / (cos^2(x) * sin^2(x)) * 1
= 1 / (cos^2(x) * sin^2(x))
Now, let's simplify the right-hand side of the equation: 1 - sin^2(x).
Since sin^2(x) + cos^2(x) = 1, we can rewrite 1 - sin^2(x) as cos^2(x):
1 - sin^2(x) = cos^2(x)
Therefore, the simplified equation becomes:
1 / (cos^2(x) * sin^2(x)) = cos^2(x)
So, the answer to the equation tan^2x(1+cot^2x) / (1-sin^2x) is cos^2(x).