1. What two common identities are most useful for simplifying 2tanx/(sin^2x + cos^2x - tan^2x)?
Two identities are: sin^2x + cos^2x = 1 and tan2x=2tanx/1-tan^2x
your are correct
2tanx/(sin^2x + cos^2x - tan^2x)
= 2tanx/(1 - tan^2 x)
= tan(2x)
Thank you~
To simplify 2tanx/(sin^2x + cos^2x - tan^2x), we can use the two common identities:
1) sin^2x + cos^2x = 1 (the Pythagorean identity)
2) tan2x = 2tanx / (1 - tan^2x)
Let's apply these identities step-by-step to simplify the given expression:
Step 1: Apply the Pythagorean identity
sin^2x + cos^2x = 1
Step 2: Substitute the result of the Pythagorean identity into the original expression
2tanx / (1 - tan^2x)
Step 3: Apply the identity tan2x = 2tanx / (1 - tan^2x)
= 2tanx / (tan2x)
Step 4: Simplify by canceling out the common factor of 2tanx
= 1 / (tan2x)
So, the two common identities most useful for simplifying 2tanx/(sin^2x + cos^2x - tan^2x) are sin^2x + cos^2x = 1 and tan2x = 2tanx / (1 - tan^2x).
To simplify the expression 2tanx/(sin^2x + cos^2x - tan^2x), we can make use of the two common identities you mentioned: sin^2x + cos^2x = 1 and tan2x = 2tanx/(1 - tan^2x).
First, let's simplify the denominator by substituting sin^2x + cos^2x with 1:
2tanx/(1 - tan^2x - tan^2x)
Next, note that the expression 1 - tan^2x - tan^2x can be further simplified as 1 - 2tan^2x. So now we have:
2tanx/(1 - 2tan^2x)
We can now use the identity tan2x = 2tanx/(1 - tan^2x) to simplify further. By rearranging this identity, we get:
2tan^2x = tan2x(1 - tan^2x)
Substituting this back into the expression, we have:
2tanx/(1 - tan^2x(1 - tan^2x))
The next step would be to expand the expression (1 - tan^2x(1 - tan^2x)):
1 - tan^2x + tan^4x
Our simplified expression becomes:
2tanx/(1 - tan^2x + tan^4x)
So, the two common identities that are most useful for simplifying 2tanx/(sin^2x + cos^2x - tan^2x) are sin^2x + cos^2x = 1 and tan2x = 2tanx/(1 - tan^2x).