Prove the identity:
sec^4x - tan^4x = 1+2tan^2x
Steps r not clear
Steps r not clear
To prove the identity: sec^4x - tan^4x = 1 + 2tan^2x, we can start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS) of the equation:
LHS: sec^4x - tan^4x
Step 1: Convert sec^2x to tan^2x using the identity: sec^2x = 1 + tan^2x
Replacing sec^2x in LHS:
= (1 + tan^2x)^2 - tan^4x
Step 2: Expand the squared expression using the binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2:
= (1 + 2tan^2x + tan^4x) - tan^4x
Step 3: Simplify the expression by canceling out the like terms (tan^4x - tan^4x):
= 1 + 2tan^2x
Now we have the RHS, which proves that sec^4x - tan^4x is equal to 1 + 2tan^2x.
Therefore, the identity has been proven.
From left side:
sec^4x - tan^4x
factors into:
(sec(x)+tan(x))*(sec(x)-tan(x))*(sec^2(x) + tan^2(x))
(sec(x)+tan(x))*(sec(x)-tan(x)) =sec^2x - tan^2x
and from trig identity:
sec^2x - tan^2x = 1
left side:
1*(sec^2(x) + tan^2(x))
Right side:
1+2tan^2(x)
from the trig identity:
sec^2x - tan^2x = 1
sec^2x - tan^2x + 2tan^2x = 1+2tan^2x
simp lying this:
sec^2x + tan^2x
So right side now matches left side.