Express the following complex numbers in the form

tan α – i

Given Complex number is tan-i

We know that the polar form of a complex number Z=x+iy is given by Z=|Z|(cosθ+isinθ)

Where,

|Z|=modulus of complex number=

θ =arg(z)=argument of complex number=

We know that tanα is a periodic function with period .

We have lying in the interval

Case1:

⇒

⇒

⇒

⇒

Since sec is positive in the interval

⇒

⇒

⇒

Since cot is positive in the interval

⇒ (∵ θ lies in 4^th quadrant)

⇒

⇒ z=sec(sin-icos)

∴ The polar form is z=sec(sin-icos)

Case2:

⇒

⇒

⇒

⇒

Since sec is negative in the interval .

⇒

⇒

⇒

Since cot is negative in the interval .

⇒ . (∵ θ lies in3^rd quadrant)

⇒

⇒ z=-sec(-sin+icos)

⇒ z=sec(sin-icos)

∴ The polar form is z=sec(sin-icos).