how do i solve ( 3tan²(x - 1) ) / ( tan³(x) - 3tan(x) )
To solve the expression (3tan²(x - 1)) / (tan³(x) - 3tan(x)), we can simplify it by using trigonometric identities.
Let's break it down step by step:
Step 1: Identify trigonometric identities
We need to recall the trigonometric identity for the square of the tangent function:
tan²(x) = sec²(x) - 1
Step 2: Simplify the numerator
Using the identity above, we can rewrite the numerator as:
3(tan²(x - 1)) = 3(sec²(x - 1) - 1)
Step 3: Simplify the denominator
Now, looking at the denominator, we can factor out a common factor of tan(x):
tan³(x) - 3tan(x) = tan(x)(tan²(x) - 3)
Using the identity we learned in Step 1, we can rewrite the denominator as:
tan(x)(sec²(x) - 2)
Step 4: Rewrite the expression
Now we can rewrite the expression using the simplified numerator and denominator:
(3(sec²(x - 1) - 1)) / (tan(x)(sec²(x) - 2))
Step 5: Simplify further if needed
At this point, we have simplified the expression as much as we can by using trigonometric identities. If you have any additional instructions or requirements, please provide them, and I can help you further.