Question

Express tan^-1 (cosx/1-sinx
the simplest form
II < x < Il in​

Answer 1

Given :  Tan⁻¹ ( Cosx / (1 - Sinx) )

To find : Simplest form

Solution:

Tan⁻¹ ( Cosx / (1 - Sinx) )

Using Cos2θ  = Cos²θ - Sin²θ

& Sin2θ  = 2SinθCosθ

Cos²θ + Sin²θ = 1

= Tan⁻¹ ( (Cos²(x/2) - Sin²(x/2)) / (Cos²(x/2) +Sin²(x/2) - 2Sin(x/2)Cos(x/2) ) )

using a² - b² = (a + b)(a - b)

& a² + b² - 2ab =  (a - b)²

=  Tan⁻¹ ( (Cos(x/2)+ Sin(x/2)(Cos(x/2) - Sin(x/2)  / (Cos(x/2) -Sin(x/2))²  )

= Tan⁻¹ ( (Cos(x/2)+ Sin(x/2) / (Cos(x/2) -Sin(x/2) ) )

Dividing numerator & denominator by Cos(x/2)

= Tan⁻¹ ( (1+ Tan(x/2) / (1 -Tan(x/2) ) )

= Tan⁻¹  (Tan (π/4 + x/2) )

=  π/4  + x/2

Tan⁻¹ ( Cosx / (1 - Sinx) ) =  π/4  + x/2

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