Use Trig identities to verify that sec^4(x)-tan^4(x)=1+2tan^2(x), Only work with one side of the equation
sec^4(x) - tan^4(x) =
= (1 + tan^2(x))^2 - tan^4(x)
= 1 + 2tan^2(x) + tan^4(x) - tan^4(x)
= 1 + 2tan^2(x)
QED
To verify that sec^4(x) - tan^4(x) is equal to 1 + 2tan^2(x), we will work with the left-hand side (LHS) of the equation only.
First, let's rewrite sec^4(x) and tan^4(x) using the trigonometric identities.
Identity 1: sec^2(x) = 1 + tan^2(x)
Using this identity, we can express sec^4(x) as follows:
sec^4(x) = (1 + tan^2(x))^2
Identity 2: tan^2(x) = sec^2(x) - 1
Using this identity, we can express tan^4(x) as follows:
tan^4(x) = (sec^2(x) - 1)^2
Now, let's substitute these expressions back into the original equation:
LHS = sec^4(x) - tan^4(x)
= (1 + tan^2(x))^2 - (sec^2(x) - 1)^2
To solve this equation, let's expand the squares:
LHS = (1 + 2tan^2(x) + tan^4(x)) - (sec^4(x) - 2sec^2(x) + 1)
Now, we can cancel out the common terms:
LHS = 2tan^2(x) + tan^4(x) - sec^4(x) + 2sec^2(x)
Next, we can rearrange the terms:
LHS = tan^4(x) - sec^4(x) + 2tan^2(x) + 2sec^2(x)
Finally, observe that tan^4(x) - sec^4(x) can be expressed as:
tan^4(x) - sec^4(x) = (tan^2(x) - sec^2(x))(tan^2(x) + sec^2(x))
Since tan^2(x) + sec^2(x) = 1 (a trigonometric identity), we can simplify the equation further:
LHS = (tan^2(x) - sec^2(x))(1 + 2tan^2(x) + 2sec^2(x))
= -1(1 + 2tan^2(x) + 2sec^2(x))
= -1 - 2tan^2(x) - 2sec^2(x)
So, the left-hand side (LHS) of the equation is indeed equal to -1 - 2tan^2(x) - 2sec^2(x).
Therefore, by using trigonometric identities, we have verified that sec^4(x) - tan^4(x) is equal to 1 + 2tan^2(x).