Vertify the identities, clearly show each step (vertify one side that they are equal to each other)
sec^2+csc^2=sec^2+csc^2
There is nothing to prove. You have exactly the same terms on both sides already.
I'd call that a tautology, not an identity
To verify the given identity, we will start with one side of the equation and simplify it until it matches the other side. Let's begin:
Starting with the left-hand side (LHS) of the equation:
LHS = sec^2 + csc^2
Now, let's work on simplifying the left-hand side:
Using the Pythagorean identity for tangent (tan^2θ + 1 = sec^2θ), we can rewrite the left-hand side as follows:
LHS = (tan^2θ + 1) + (1 + cot^2θ)
Expanding this expression, we get:
LHS = tan^2θ + 1 + 1 + cot^2θ
Next, we can simplify the terms involving tangent and cotangent.
Using the Pythagorean identity for cotangent (1 + cot^2θ = csc^2θ), we can substitute it into our expression:
LHS = tan^2θ + 1 + csc^2θ
Now, let's focus on the right-hand side (RHS) of the equation:
RHS = sec^2θ + csc^2θ
We can see that the right-hand side is already in the same form as our expression for the left-hand side. Therefore, the right-hand side (RHS) is equal to:
RHS = sec^2θ + csc^2θ
Since we have obtained the same expression for both sides of the equation, we can conclude that:
LHS = RHS
Hence, the given identity is verified.