simplify (1-cot^2x sec^2x) / ( cot^2x)
And show work please!
I usually change all I can to sines and cosines.
Have you tried that ?
I did but I got lost :(
In this case, just divide through by cot^2 to get
tan^2 - sec^2 = -1
Wow I feel stupid. Lol thanks tho!
We've all been there.
Good judgment comes from experience.
Experience comes from bad judgment.
To simplify the expression (1 - cot^2x sec^2x) / cot^2x, let's break it down step by step:
Step 1: Rewrite sec^2x in terms of sin^2x and cos^2x.
Recall that sec^2x = 1/cos^2x. Therefore, we can replace sec^2x in the expression with 1/cos^2x.
(1 - cot^2x (1/cos^2x)) / cot^2x
Step 2: Simplify the expression inside the brackets.
To simplify the expression (1 - cot^2x (1/cos^2x)), we can first rewrite cot^2x as 1/tan^2x.
(1 - (1/tan^2x) (1/cos^2x)) / cot^2x
Step 3: Simplify the product of fractions.
To simplify the expression, multiply the fractions in the numerator.
(1 - (1/(tan^2x * cos^2x))) / cot^2x
Step 4: Simplify the denominator.
Since cot^2x is equal to 1/tan^2x, we can replace cot^2x in the expression with 1/tan^2x.
(1 - (1/(tan^2x * cos^2x))) / (1/tan^2x)
Step 5: Simplify further.
To simplify further, combine the fractions in the numerator.
(1 - 1/(tan^2x * cos^2x))/(1/tan^2x)
Step 6: Multiply the numerator by the reciprocal of the denominator.
To divide by 1/tan^2x, we can multiply the numerator by tan^2x.
[(1 - 1/(tan^2x * cos^2x))/(1)] * tan^2x
Step 7: Simplify the numerator and cancel out the common terms.
Apply the distributive property to multiply tan^2x inside the brackets.
[(tan^2x - 1/(cos^2x))/(1)] * tan^2x
tan^2x * (tan^2x - 1/(cos^2x))
Step 8: Simplify further if desired.
If you want to simplify the expression even further, you can expand and rewrite the terms in terms of sinx, cosx, and tanx.
tan^2x - \[\frac{1}{cos^2x}\]
=(sin^2x/cos^2x) - \[\frac{1}{cos^2x}\]
=(sin^2x - 1)/cos^2x
So, the simplified expression is (sin^2x - 1)/cos^2x.