Question

Simplify the expression: (1-cosx)(1+cosx)/sinx

Answer 1

set a = 1 and b = cos(x)

then the expression becomes:

\[\frac{ (a-b)(a+b) }{ \sin(x) }\]

which equals:
\[\frac{ a^2-b^2 }{ \sin(x) }\]

now plug back in a and b:
\[\frac{ 1-\cos^2(x) }{ \sin(x) }\]

now you know the identity:
\[\cos^2(x)+\sin^2(x)=1\]

so:\[1-\cos^2(x)=\sin^2(x)\]

so if you plug that in to your expression you get:
\[\frac{ \sin^2(x) }{ \sin(x) }\]


which simply equals:\[\sin(x)\]



ANSWER: \[\sin(x)\]