Question

$\rm{Question:-}$
If cosec θ-sin θ=l and sec θ- cos θ=m, prove that$\tt{l^{2} m^{2} (l^{2} +m^{2} +3)=1}$

Answer 1

✰ Some important ratios & identities ✰

Reciprocal Identities :

➺ Sinθ = 1/Cosecθ

➺ Cosθ = 1/Secθ

➺ Tanθ = 1/Cotθ

Qoutient Identities :

➺ Sinθ/Cosθ = Tanθ

➺ Cosθ/Sinθ = Cotθ

Pythagorean Identities :

➺ Cos²θ + Sin²θ = 1

➺ Tan²θ + 1 = Sec²θ

➺ 1 + Cot²θ = Cosec²θ

Cofunction Identities :

➺ Sinθ = Cos (90° – θ)

➺ Tanθ = Cot (90° – θ)

➺ Secθ = Cosec (90° – θ)

Opposite angle Identities :

➺ Sin (–A) = – SinA

➺ Cos (–A) = CosA

➺ Tan (–A) = – TanA

Sum and difference Identities :

➺ Sin (α ± β) = SinαCosβ ± CosαSinβ

➺ Cos (α ± β) = CosαCosβ ± SinαSinβ

➺ Tan (α ± β) = Tanα ± Tanβ/1 ± TanαTanβ

Double angle Identities :

➺ Sin 2θ = 2 SinθCosθ

➺ Cos 2θ = Cos²θ – Sin²θ

➺ Cos 2θ = 2 Cos²θ – 1

➺ Cos 2θ = 1 – 2 Sin²θ

➺ Tan 2θ = 2 Tanθ/1 – Tan²θ

Product sum Identities :

➺ SinαCosβ = ½ Sin (α + β) + Sin (α – β)

➺ CosαSinβ = ½ Sin (α + β) – Sin (α – β)

➺ SinαSinβ = ½ Cos (α – β) – Cos(α + β)

➺ CosαCosβ = ½ Cos (α + β) + Cos (α–β)