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Mathematics

On simplifying the expression tan3 θ−1tan θ−1\dfrac{\text{tan}^3 \text{ θ} - 1}{\text{tan θ} - 1}, we get :

  1. sec2 θ + tan θ

  2. sec2 θ - tan θ

  3. sec θ + tan2 θ

  4. sec θ - tan2 θ

Trigonometric Identities

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Answer

Solving,

⇒tan3 θ−1tan θ−1⇒tan3 θ−13tan θ−1⇒(tan θ−1)(tan2 θ+tan θ+12)tan θ−1⇒tan2 θ+tan  θ + 1⇒sec2 θ−1+tan  θ + 1⇒sec2 θ+tan  θ.\Rightarrow \dfrac{\text{tan}^3 \text{ θ} - 1}{\text{tan} \text{ θ} - 1} \\[1em] \Rightarrow \dfrac{\text{tan}^3 \text{ θ} - 1^3}{\text{tan} \text{ θ} - 1} \\[1em] \Rightarrow \dfrac{(\text{tan} \text{ θ} - 1)(\text{tan}^2 \text{ θ} + \text{tan θ} + 1^2)}{\text{tan θ} - 1} \\[1em] \Rightarrow \text{tan}^2 \text{ θ} + \text{tan \text{ θ} + 1} \\[1em] \Rightarrow \text{sec}^2 \text{ θ} - 1 + \text{tan \text{ θ} + 1} \\[1em] \Rightarrow \text{sec}^2 \text{ θ} + \text{tan \text{ θ}}.

Hence, Option 1 is the correct option.

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