# Prove this trigonometric identity:?

cot(A) cos (A) / csc^2(A)-1 = sin(A)

I'm sorry. I've been trying to figure this out for an hour; and, it is beginning to really bother me. I keep getting my answer down to -1/sin(A) and I know that isn't right. Please help!

### 2 Answers

- ?Lv 46 years agoFavorite Answer
cot(A) cos(A) / csc²(A) - 1 = sin(A)

We solve the Left Hand Side to get the Right Hand Side.

(Left Hand Side) LHS

= cot(A) cos(A) / csc²(A) - 1

= [ cos(A) / sin(A) ] cos(A) / [ { 1/sin²(A) } - 1 ] ................... Since [ cot x = cos x / sinx ] & [ csc²x = 1 / sin²x ]

= [ cos²(A) / sin(A) ] / [ { 1-sin²(A) } / sin²(A) ] ....... Since [ cosA*cosA = cos²A ] & [ (1/x) - 1 ] = [ (1-x) / x ]

= [cos²(A) / 1 ] / [ cos²(A) / sin(A) ] ......Cut the first Den. sin(A) with the 2nd Den. sin²(A) & [ 1-sin²x = cos²x]

= [cos²(A) / 1 ] * [ sin(A) / cos²(A) ] ...... Since for example [ (1/x) / (2/x) ] = [ (1/x) * (x/2) ] = 1/2

= sin(A) / 1 ......... Cut the cos²(A) Nr. & Dr.

= sin(A) = Right Hand Side (RHS)

Hence proved

cot(A) cos(A) / csc²(A)-1 = sin(A)

- Anonymous6 years ago
do ur own hmw

Welcome!