Simplify (1 + tanθ + secθ) (1 + cotθ - cosecθ)
Answer:
2
- We need to find following product
S = (1 + tanθ + secθ) (1 + cotθ - cosecθ) - 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) - Using identities cotθ tanθ = 1, cotθ secθ = cosecθ and cosecθ tanθ = secθ
⇒ S = (1 + tanθ + secθ) + (cotθ + 1 + cosecθ ) - (cosecθ + secθ + cosecθ secθ)
- 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 - Using identities tanθ = sinθ/cosθ, and cotθ = cosθ/sinθ
⇒ S = 2 +
+sinθ cosθ
- cosecθ secθcosθ sinθ
⇒ S = 2 +
- cosecθ secθsin2θ + cos2θ sinθcosθ
⇒ S = 2 +
- cosecθ secθ1 sinθcosθ
⇒ S = 2 + cosecθ secθ - cosecθ secθ
⇒ S = 2