Question

Prove the identity sin(2x)/sinx-cos(2x)/cosx= secx​

Answer 1

Answer:

$\frac{ \sin(2x) }{ \sin(x) } - \frac{ \cos(2x) }{ \cos(x) } \\ = \frac{2 \sin(x) \cos(x) }{ \sin(x) } - \frac{ { \cos(x) }^{2} - { \sin(x) }^{2} }{ \ \cos(x) } \\ = 2 \cos(x) - \frac{ { \cos(x) }^{2} - { \sin(x) }^{2} }{ \cos(x) } \\ = \frac{2 { \cos(x) }^{2} - { \cos(x) }^{2} + { \sin(x) }^{2} }{ \cos(x) } \\ = \frac{ { \cos(x) }^{2} + { \sin(x) }^{2} }{ \cos(x) } \\ = \frac{1}{ \cos(x) } \\ = \sec(x)$

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