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QuestionMultiply and simplify: (2−tan⁡θ)(2+tan⁡θ)(2-\tan \theta)(2+\tan \theta).

Studdy Solution

STEP 1

Assumptions1. We are given the expression (−tan⁡θ)(+tan⁡θ)(-\tan \theta)(+\tan \theta). We are asked to simplify this expression3. We will use the identity tan⁡θ+1=sec⁡θ\tan^ \theta +1 = \sec^ \theta in the simplification process

STEP 2

We will use the difference of squares formula to simplify the expression. The difference of squares formula is a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b).
(2−tan⁡θ)(2+tan⁡θ)=22−(tan⁡θ)2(2-\tan \theta)(2+\tan \theta) =2^2 - (\tan \theta)^2

STEP 3

Now, simplify the right-hand side of the equation.
22−(tan⁡θ)2=−tan⁡2θ2^2 - (\tan \theta)^2 = - \tan^2 \theta

STEP 4

Now, we will use the identity tan⁡2θ+1=sec⁡2θ\tan^2 \theta +1 = \sec^2 \theta to replace tan⁡2θ\tan^2 \theta.
4−tan⁡2θ=4−(sec⁡2θ−1)4 - \tan^2 \theta =4 - (\sec^2 \theta -1)

STEP 5

implify the right-hand side of the equation.
4−(sec⁡2θ−1)=5−sec⁡2θ4 - (\sec^2 \theta -1) =5 - \sec^2 \thetaSo, the simplified form of (2−tan⁡θ)(2+tan⁡θ)(2-\tan \theta)(2+\tan \theta) is 5−sec⁡2θ5 - \sec^2 \theta.

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