Questions: Simplify the expression. (tan x + sec x)(tan x - sec x) (tan x + sec x)(tan x - sec x) = (Use integers or decimals for any numbers in the expression.)

Solution Steps

To simplify the given expression \((\tan x + \sec x)(\tan x - \sec x)\), we can use the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \tan x\) and \(b = \sec x\). Therefore, the expression simplifies to \(\tan^2 x - \sec^2 x\).

To simplify the expression \((\tan x + \sec x)(\tan x - \sec x)\), we recognize that it follows the difference of squares pattern, which is given by: \[ (a + b)(a - b) = a^2 - b^2 \] Here, let \(a = \tan x\) and \(b = \sec x\). Thus, we can rewrite the expression as: \[ \tan^2 x - \sec^2 x \]

We know from trigonometric identities that: \[ \sec^2 x = 1 + \tan^2 x \] Substituting this into our expression gives: \[ \tan^2 x - \sec^2 x = \tan^2 x - (1 + \tan^2 x) = \tan^2 x - 1 - \tan^2 x \] This simplifies to: \[ -1 \]

Final Answer