Questions: sec^2 theta + cot^2 theta = csc^2 theta sec^2 theta - 1

Solution Steps

To verify if the given equation is a trigonometric identity, we can simplify both sides of the equation using known trigonometric identities and see if they are equivalent.

We start with the given trigonometric equation: \[ \sec^2 \theta + \cot^2 \theta = \csc^2 \theta \sec^2 \theta - 1 \]

The left-hand side of the equation is: \[ \sec^2 \theta + \cot^2 \theta \]

The right-hand side of the equation is: \[ \csc^2 \theta \sec^2 \theta - 1 \]

After simplifying both sides, we get: \[ \text{LHS} = \cot^2 \theta + \sec^2 \theta \] \[ \text{RHS} = -1 - \frac{8}{\cos(4\theta) - 1} \]

We need to check if the simplified forms of LHS and RHS are equivalent. From the output, we see that: \[ \cot^2 \theta + \sec^2 \theta \neq -1 - \frac{8}{\cos(4\theta) - 1} \]

Since the simplified forms of LHS and RHS are not equal, the given equation is not a trigonometric identity.

Final Answer