Question

prove tan x-cot x/sin x(cos x)=sec^2x- csc^2x

Answer 1

Answer:

$\frac{ \tan(x) - \cot(x) }{ \sin(x) \cos(x) } = { \sec(x) }^{2} - \csc(x) {}^{2} \\ lhs \: = \frac{ \tan(x) - \cot(x) }{ \sin(x) \cos(x) } \\ \: \: = \frac{ \tan(x) }{ \sin(x) \cos(x) } - \frac{ \cot(x) }{ \sin(x) \cos(x) } \\ \: \: = \frac{1}{ { \cos(x) }^{2} } - \frac{1}{ { \sin(x) }^{2} } \\ \: \: \: = { \sec(x) }^{2} - { \csc(x) }^{2} \\ \: \: \: = rhs \: \\ proved$