Verify the identities by transforming the left-hand side into the right-hand side. Show work.
(tan^2X)/(1-cos^2x)= sec^2x
HELP
LS = tan^2 x/(sin^2 x)
= (sin^2 x)/cos^2 x) ( sin^2 x)
= 1/cos^2 x
= sec^2 x
= RS
To verify the given identity and transform the left-hand side (LHS) into the right-hand side (RHS), we can use the trigonometric identity:
1 - cos^2(x) = sin^2(x)
We can start by manipulating the LHS:
(tan^2(x))/(1 - cos^2(x))
Now, using the property of tan^2(x) = sec^2(x) - 1:
(sec^2(x) - 1) / (1 - cos^2(x))
Next, since sec^2(x) = 1 + tan^2(x):
((1 + tan^2(x)) - 1) / (1 - cos^2(x))
Simplifying further:
tan^2(x) / (1 - cos^2(x))
Now, substituting sin^2(x) for 1 - cos^2(x):
tan^2(x) / sin^2(x)
Finally, using the identity tan(x) = sin(x)/cos(x):
(sin^2(x)/cos^2(x)) / sin^2(x)
Cancel out the common factor:
1/cos^2(x)
This simplifies to:
sec^2(x)
Therefore, the LHS is equal to the RHS, and the identity has been verified.