Prove that the equation Is an identity.
Sec^4x - Tan^4x = Sec^2x + Tan^2x
LS = (sec^2 x - tan^2x)(sec^2x + tan^2x)
= (1 + tan^2x - tan^2x)(sec^2x + tan^2x)
= 1(sec^2x + tan^2x)
= RS
To prove that the given equation, Sec^4x - Tan^4x = Sec^2x + Tan^2x, is an identity, we need to simplify both sides of the equation and show that they are equal to each other.
Starting from the left-hand side (LHS):
LHS = Sec^4x - Tan^4x
Using the identity:
Sec^2x = 1 + Tan^2x
We can rewrite the equation as:
LHS = (Sec^2x)^2 - Tan^4x
Expanding the expression:
LHS = (1 + Tan^2x)^2 - Tan^4x
Simplifying:
LHS = 1 + 2(Tan^2x) + (Tan^2x)^2 - Tan^4x
Next, let's simplify the right-hand side (RHS):
RHS = Sec^2x + Tan^2x
Using the same identity as before:
Sec^2x = 1 + Tan^2x
RHS = 1 + Tan^2x + Tan^2x
RHS = 1 + 2(Tan^2x)
Now, if we compare the simplified versions of the LHS and RHS, we can see that they are equal:
LHS = 1 + 2(Tan^2x) + (Tan^2x)^2 - Tan^4x
RHS = 1 + 2(Tan^2x)
After re-arranging and simplifying the terms, we can see that the LHS is equal to the RHS. Therefore, the given equation Sec^4x - Tan^4x = Sec^2x + Tan^2x is indeed an identity.
To prove that the equation is an identity, we need to show that the equation holds true for all values of x.
Starting with the left-hand side (LHS) of the equation, we have:
LHS = sec^4(x) - tan^4(x)
Using the trigonometric identity: sec^2(x) - tan^2(x) = 1, we can rewrite the equation as:
LHS = (sec^2(x))^2 - (tan^2(x))^2
Now, let's factor the expression:
LHS = (sec^2(x) + tan^2(x)) * (sec^2(x) - tan^2(x))
Using the trigonometric identity again: sec^2(x) - tan^2(x) = 1, we can simplify further:
LHS = (sec^2(x) + tan^2(x)) * 1
Since the trigonometric identity sec^2(x) + tan^2(x) = 1, we can substitute this value into the equation:
LHS = 1 * 1
LHS = 1
Now, let's simplify the right-hand side (RHS) of the equation:
RHS = sec^2(x) + tan^2(x)
Since the trigonometric identity sec^2(x) + tan^2(x) = 1, the RHS also simplifies to:
RHS = 1
Comparing the LHS and RHS of the equation, we can see that they are both equal to 1, which means that the original equation is an identity.
Therefore, the equation sec^4(x) - tan^4(x) = sec^2(x) + tan^2(x) is an identity.