Verify (csc^2x-1)/cos^2x=csc^2x.
After making (csc^2x-1) into ((1/sin^2x)-1) and cos^2 into 1-sin^2x, what do I do next??
I did it wrong, nevermind.
One of the identities you should know is
cot^2 x = csc^2 x - 1
then ...
LS = (csc^2x-1)/cos^2x
= cot^2 x / cos^2 x
= (cos^2 x/sin^2 x) / cos^2 x
= 1/sin^2 x
= csc^2 x
= RS
After expressing (csc^2x-1) as ((1/sin^2x)-1) and cos^2x as 1-sin^2x, the next step is to simplify the equation.
Let's substitute these expressions back into the original equation:
((1/sin^2x)-1)/((1-sin^2x)/1) = (1/sin^2x) [cosec^2x is equivalent to (1/sin^2x)]
Now, simplify the left-hand side of the equation by using the division of fractions rule. In this case, you need to multiply the numerator by the reciprocal of the denominator:
((1/sin^2x)-1)*(1/((1-sin^2x)/1))
Next, multiply the numerator by the reciprocal of the denominator:
((1/sin^2x)-1)*(1*(1/ (1-sin^2x)))
Now, multiply the numerators:
(1/sin^2x)*(1/ (1-sin^2x)) - 1 * 1/ (1-sin^2x)
Simplify each term:
1/(sin^2x*(1-sin^2x)) - 1/(1-sin^2x)
Now, observe that sin^2x*(1-sin^2x) can be simplified:
sin^2x*(1-sin^2x) = sin^2x-sin^4x
Substitute this back into the equation:
1/(sin^2x-sin^4x) - 1/(1-sin^2x)
To combine the two terms over a common denominator, you multiply the first fraction by (1-sin^2x)/(1-sin^2x):
(1/(sin^2x-sin^4x))*( (1-sin^2x)/(1-sin^2x) )
Simplify the numerator:
(1*(1-sin^2x))/(sin^2x*(1-sin^2x))
Now, simplify the denominator:
sin^2x*(1-sin^2x) = sin^2x-sin^4x
1/(sin^2x-sin^4x)*(1-sin^2x)/(sin^2x*(1-sin^2x))
Now, you can see that the (sin^2x - sin^4x) terms cancel out:
1/(sin^2x) * (1-sin^2x)/(sin^2x)
Simplify:
(1-sin^2x)/(sin^2x)
Now, recall that (1-sin^2x) is equal to cos^2x:
cos^2x / sin^2x
Finally, using the quotient identity (cos^2x = 1/csc^2x):
1/csc^2x
Therefore, we have verified that (csc^2x-1)/cos^2x is equal to csc^2x.