Questions: Prove the identity sec^4(x) - sec^2(x) = tan^2(x) + tan^4(x)

Solution Steps

To prove the identity \(\sec^4(x) - \sec^2(x) = \tan^2(x) + \tan^4(x)\), we can start by manipulating the left-hand side of the equation.

1. Factor out \(\sec^2(x)\) from the left-hand side.
2. Use the Pythagorean identity \(\sec^2(x) = 1 + \tan^2(x)\) to simplify the expression.
3. Substitute \(\sec^2(x)\) with \(1 + \tan^2(x)\) and simplify to show that both sides are equal.

We start with the left-hand side of the identity: \[ \sec^4(x) - \sec^2(x) \] We factor out \(\sec^2(x)\): \[ \sec^2(x) (\sec^2(x) - 1) \]

We use the Pythagorean identity \(\sec^2(x) = 1 + \tan^2(x)\) to simplify the expression: \[ \sec^2(x) (1 + \tan^2(x) - 1) = \sec^2(x) \tan^2(x) \]

Substitute \(\sec^2(x) = 1 + \tan^2(x)\) into the expression: \[ (1 + \tan^2(x)) \tan^2(x) \] Expand the expression: \[ \tan^2(x) + \tan^4(x) \]

The right-hand side of the original identity is: \[ \tan^2(x) + \tan^4(x) \] Since both sides are equal, the identity is proven.

Final Answer