Questions: Verify each identity. 1. (tan^2 x+1)(cos^2 x-1)=-tan^2 x

Solution Steps

To verify the trigonometric identity \(\left(\tan ^{2} x+1\right)\left(\cos ^{2} x-1\right)=-\tan ^{2} x\), we can use known trigonometric identities. Specifically, we can use the Pythagorean identity \(\tan^2 x + 1 = \sec^2 x\) and the identity \(\cos^2 x - 1 = -\sin^2 x\). By substituting these identities into the given expression, we can simplify and verify the identity.

1. Substitute \(\tan^2 x + 1\) with \(\sec^2 x\).
2. Substitute \(\cos^2 x - 1\) with \(-\sin^2 x\).
3. Simplify the expression to verify if it equals \(-\tan^2 x\).

We start with the left-hand side of the identity: \[ \text{LHS} = \left(\tan^2 x + 1\right)\left(\cos^2 x - 1\right) \]

Using the Pythagorean identity, we know: \[ \tan^2 x + 1 = \sec^2 x \] And we also know: \[ \cos^2 x - 1 = -\sin^2 x \] Substituting these into the left-hand side gives: \[ \text{LHS} = \sec^2 x \cdot (-\sin^2 x) \]

Now, we simplify the expression: \[ \text{LHS} = -\sec^2 x \cdot \sin^2 x \] Using the identity \(\sec^2 x = \frac{1}{\cos^2 x}\), we can rewrite it as: \[ \text{LHS} = -\frac{\sin^2 x}{\cos^2 x} = -\tan^2 x \]

The right-hand side of the identity is: \[ \text{RHS} = -\tan^2 x \]

We have shown that: \[ \text{LHS} = -\tan^2 x \] and \[ \text{RHS} = -\tan^2 x \] Since both sides are equal, the identity is verified.

Final Answer