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Sticky Note
Cos2θ/Cos⁴θ-Sin⁴θ = _____?
  1. -1
  2. 0
  3. 1
  4. None of these
Explanation

Let's simplify the expression:

cos⁡2θcos⁡4θ−sin⁡4θ

Step 1: Simplify the denominator
Note that:

cos⁡4θ−sin⁡4θ=(cos⁡2θ)2−(sin⁡2θ)2=(cos⁡2θ−sin⁡2θ)(cos⁡2θ+sin⁡2θ)

Since cos⁡2θ+sin⁡2θ=1, we have:

cos⁡4θ−sin⁡4θ=(cos⁡2θ−sin⁡2θ)⋅1=cos⁡2θ

(because cos⁡2θ=cos⁡2θ−sin⁡2θ).

Step 2: Substitute back into the expression

cos⁡2θcos⁡4θ−sin⁡4θ=cos⁡2θcos⁡2θ=1

Therefore, the expression simplifies to 1.

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