Prove:
(tan x+sec x)^2 = 2sec^2(x)+2tan(x)secx-1
To prove the given equation, (tan x + sec x)^2 = 2sec^2(x) + 2tan(x)sec(x) - 1, we'll start by simplifying both sides individually.
Let's simplify the left-hand side (LHS) first:
(tan x + sec x)^2
Expanding the square, we get:
(tan x)^2 + 2tan(x)sec(x) + (sec x)^2
Next, let's simplify the right-hand side (RHS):
2sec^2(x) + 2tan(x)sec(x) - 1
Now, we will combine like terms on the right-hand side:
2sec^2(x) + 2tan(x)sec(x) - 1
Since 2tan(x)sec(x) is already present on the LHS, we can rewrite the right-hand side as:
(tan x)^2 + 2tan(x)sec(x) + (sec x)^2 - 1
Now, both sides of the equation match:
LHS = (tan x)^2 + 2tan(x)sec(x) + (sec x)^2
RHS = (tan x)^2 + 2tan(x)sec(x) + (sec x)^2 - 1
Since both sides of the equation are equal, we have proven:
(tan x + sec x)^2 = 2sec^2(x) + 2tan(x)sec(x) - 1.