Question

prove that sinA/1-cosA=cosecA+cotA ​

Answer 1

Step-by-step explanation:

R.T.P : sinA / 1 - cosA = cosecA + cotA

⇒ $\frac{sinA}{1 - cosA}$ × $\frac{1 + cosA}{1 + cosA}$

= $\frac{sinA( 1 + cosA ) }{1 - cos^2A}$

= $\frac{sinA ( 1 + cosA)}{sin^2A}$  by $sin^2A = 1 - cos^2A$

= $\frac{1 + cosA}{sinA}$

= $\frac{1}{sinA} + \frac{cosA}{sinA}$

= cosecA + cotA  as cosecA = 1/sinA & cotA = cosA/sinA

Answer 2

Answer:

Proved by solving equation sinA/1-cosA = cosecA+cotA

Step-by-step explanation:

Here given that

sinA/1-cosA = cosecA+cotA

Let solve Right hand side of given eqation

RHS

cosecA + cotA = (1/sinA) + (cosA/sinA)

               (Because cosecA=1/sinA    and cotA=cosA/sinA)

                   Divide  (1/sinA) + (cosA/sinA) by sinA, We get

                 = (1+cosA)/sinA

  Divide numerator and denominator by (1-cosA), We get

                 =  (1+cosA)*(1-cosA)/sinA*(1-cosA)

  But We know that (x+y)( x-y)=x²-y²

                 = (1-cos²A)/sinA*(1-cosA)..............................(1)

        But we know the identity  sin²A+cos²A = 1

                          therefore sin²A = 1-cos²A

                 Put the value of  sin²A = 1-cos²A in equation (1)

                     = sin²A/sinA*(1-cosA)

                     = sinA/(1-cosA)  

This is Left hand side of given equation

Therefore LHS=RHS

Hence proved.