Question

Prove that sinx /1-cosx =cosecx +cotx

Answer 1

To prove---> Sinx / (1-Cosx) = Cosecx + Cotx

Proof--->

RHS= Cosecx + Cotx

We have two formulee

Cosecx = 1 / Sinx , Cotx = Cosx / Sinx , applying these formulee here, we get

= 1 / Sinx + Cosx / Sinx

Taking LCM as Sinx

= 1 + Cosx / Sinx

Multiplying by conjugate of numerator , in the numerator and denominator which is ( 1 - Cosx ) , we get

= ( 1 + Cosx ) ( 1 - Cosx ) / Sinx ( 1 - Cosx )

We have an identity ,

a² - b² = ( a + b ) ( a - b ) , applying it here we get,

= ( 1 )² - ( Cosx )² / Sinx ( 1 - Cosx )

= ( 1 - Cos²x ) / Sinx ( 1 - Cosx )

We know that , 1 - Cos²x = Sin²x , applying it here , we get

= Sin²x / Sinx ( 1 - Cosx )

Sinx cancel out from numerator and denominator and we get LHS

= Sinx / ( 1 - Cosx ) = LHS

Answer 2

Answer:

Step-by-step explanation:

Left Hand Side:

= $\frac{sinx}{1-cosx}({\frac{1+cosx}{1+cosx} )$

multiply by the conjugate

=  $\frac{sinx+sinxcosx}{1-cos²x}$

-distribute

=  $\frac{sinx}{sin^{2} x} +\frac{sincosx}{sin^{2}x }$

-use property  

$sin^{2} x+cos^{2} x=1$

$\frac{1}{sinx} +\frac{cosx}{sinx}$

= $cscx+cotx$

Right Hand Side

                                         $\mathfrak{K_ee_pS_mi_ll_in_G}$