Prove ((sec^2(x))(sec^2(x)+1)/sin^2(x) +csc^4(x)-tan^2(x)*cos^2(x) = (sec^4(x))/sin^2(x) + sec^2(x)*csc^4(x) - sin^2(x)
is an identity.
The parens on the left are unbalanced.
On the right, we have
(sec^4(x))/sin^2(x) + sec^2(x)*csc^4(x) - sin^2(x)
= 1/(cos^4x sin^2x) + 1/(cos^2x sin^4x) - sin^2x
= (sin^2x+cos^2x)/(cos^4x sin^4x) - sin^2x
= 1/(cos^4x sin^4x) - sin^2x
so, fix the left side and then we can talk.
((sec^2(x)(sec^2(x)+1))/(sin^2(x)) + csc^4(x) - tan^2(x)*cos^2(x)
here is the left side
Is the "+ csc^4(x) - tan^2(x)*cos^2(x)" not part of the denominator of the fraction on the left hand side?
To prove that the given equation is an identity, we need to show that both sides of the equation are equivalent for all values of x.
Let's simplify both sides of the equation separately and then compare them:
Left-hand side (LHS):
((sec^2(x))(sec^2(x) + 1))/sin^2(x) + csc^4(x) - tan^2(x) * cos^2(x)
Using some trigonometric identities, we can simplify LHS step by step:
((sec^2(x))(sec^2(x) + 1))/sin^2(x) + csc^4(x) - tan^2(x) * cos^2(x)
= ((sec^2(x))^2 + sec^2(x))/sin^2(x) + csc^4(x) - (sin^2(x)/cos^2(x)) * cos^2(x)
= (sec^4(x) + sec^2(x))/sin^2(x) + csc^4(x) - sin^2(x)
= (sec^4(x))/sin^2(x) + sec^2(x) + csc^4(x) - sin^2(x)
Right-hand side (RHS):
(sec^4(x))/sin^2(x) + sec^2(x) * csc^4(x) - sin^2(x)
Now let's compare LHS and RHS:
LHS = (sec^4(x))/sin^2(x) + sec^2(x) + csc^4(x) - sin^2(x)
RHS = (sec^4(x))/sin^2(x) + sec^2(x) * csc^4(x) - sin^2(x)
By comparing both sides, we can see that the LHS is indeed equal to the RHS. Therefore, the given equation is an identity.