Final answer:
To verify the identity, we can simplify both sides of the equation and check if they are equal. On the left side, we simplify the expression by multiplying the numerator and denominator by cosθ. On the right side, we expand the squared term and simplify further. After simplifying both sides, we see that they are equal, confirming the identity.
Step-by-step explanation:
To verify the identity, we can start by simplifying both sides of the equation. On the left side, we have (tanθ-secθ)/(tanθ+secθ). We can rewrite secθ as 1/cosθ. So, the left side becomes ((tanθ-1/cosθ)/(tanθ+1/cosθ)). To simplify this, multiply the numerator and denominator by cosθ to get ((tanθcosθ-1)/(tanθcosθ+1)).
On the right side, we have -(tanθ-secθ)². We can substitute secθ as 1/cosθ. So, the right side becomes -((tanθ-1/cosθ)²). Expanding this, we get -((tanθ²-2tanθ/cosθ+1/cos²θ)).
Now, we can simplify both sides further and see if they are equal.
Simplifying the left side: ((tanθcosθ-1)/(tanθcosθ+1)) = ((sinθ/cosθ)cosθ-1)/((sinθ/cosθ)cosθ+1) = (sinθ-1)/(sinθ+1).
Simplifying the right side: -((tanθ²-2tanθ/cosθ+1/cos²θ)) = -((sin²θ/cos²θ-2sinθ/cosθ+1/cos²θ)) = -(sin²θ-2sinθcosθ+1)/cos²θ = (-sin²θ+2sinθcosθ-1)/cos²θ.
After simplifying both sides, we see that (sinθ-1)/(sinθ+1) = (-sin²θ+2sinθcosθ-1)/cos²θ, which means the identity is verified.