Prove that the following is an identity

Question

( cos x + sinx )²

1 + 2 sinx cos x = cos x tan x  csc x

Answer 1

(cosx+sinx)²/(1+2sinxcosx)=cosxtanxcscx

(cos²x+2sinxcosx+sin²x)/(1+2sinxcosx)=cosxtanxcscx

cos²x+sin²x=1

(1+2sinxcosx)/(1+2sinxcosx)=cosxtanxcscx

1=cosxtanxcscx

1=cosx(sinx/cosx)(1/sinx)

cancel the 2 cosx's first

1=(sinx)(1/sinx)

1=1

 

Answer 2

cosx tanx cscx = cosx (sinx/cosx)(1/sinx) = 1

 

(cosx  +sinx)² = cos²x + 2sinx cosx +sin²x

1 + 2sinxcosx              1  + 2sinx cosx

 

                          = 1 + 2sinx cosx   = 1

                             1 + 2sinx cosx

 

Since both sides of the given equation simplify to the same expression (1), the given equation is an identity.