Simplify (1 + tanθ + secθ) (1 + cotθ - cosecθ)


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Answer:

2

Step by Step Explanation:
  1. We need to find following product
    S = (1 + tanθ + secθ) (1 + cotθ - cosecθ)
  2. On multiplying each terms
    ⇒ S = 1 (1 + tanθ + secθ) + cotθ (1 + tanθ + secθ) - cosecθ (1 + tanθ + secθ)
    ⇒ S = (1 + tanθ + secθ) + (cotθ + cotθ tanθ + cotθ secθ ) - (cosecθ + cosecθ tanθ + cosecθ secθ)
  3. Using identities cotθ tanθ = 1, cotθ secθ = cosecθ and cosecθ tanθ = secθ
    ⇒ S = (1 + tanθ + secθ) + (cotθ + 1 + cosecθ ) - (cosecθ + secθ + cosecθ secθ)
  4. Now positive cosecθ and secθ will cancel each other
    ⇒ S = (1 + tanθ + secθ) + (cotθ + 1 + cosecθ ) - (cosecθ + secθ + cosecθ secθ)
    ⇒ S = 2 + tanθ + cotθ - cosecθ secθ
  5. Using identities tanθ = sinθ/cosθ, and cotθ = cosθ/sinθ
    ⇒ S = 2 +  
    sinθ
    cosθ
      +  
    cosθ
    sinθ
      - cosecθ secθ
    ⇒ S = 2 +  
    sin2θ + cos2θ
    sinθcosθ
      - cosecθ secθ
    ⇒ S = 2 +  
    1
    sinθcosθ
      - cosecθ secθ
    ⇒ S = 2 + cosecθ secθ - cosecθ secθ
    ⇒ S = 2

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