Question

Help proving this identity step-by-step? And give reasons please? Will give medal & fan.
(sin(2x)/sin(x)) - (cos(2x)/cos(x)) = sec(x)

Answer 1

Does sin(2x)/sin(x) = sin(x)?

Answer 2

Oh wait no that would be sin^2 x...

Answer 3

sin (2x) = 2sinx cos x--> the first term becomes?

Answer 4

(2sin(x)cos(x)/sin(x)) - (cos(2x)/cos(x)

Answer 5

2cos(x) - (cos(2x)/cos(x)) ?

Answer 6

now, the second term : cos (2x) = 2cos^2 x -1, so that it is \(\dfrac{2cos^2x-1}{cosx}=\dfrac{2cos^2x}{cosx}-\dfrac{1}{cosx}\)
Simplify to get \(2cosx -\dfrac{1}{cosx}\)
Now, combine with the first term :
\[2cosx - 2cos x +\dfrac{1}{cosx}= \dfrac{1}{cos x}= secx\]

Answer 7

The key is memorize the identities. :)

Answer 8

Ohhh okay (: I see, thank you very much!