Question

How do you prove `sin x + cos x * cot x = csc x`?

Answer 1

We will use the following identities to attack the problem:

•`cotx = 1/tanx = 1/(sinx/cosx) = cosx/sinx`
•`cscx = 1/sinx`
•`cos^2x + sin^2x = 1`

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`sinx + cosx * cosx/sinx = 1/sinx`

`sinx + cos^2x/sinx = 1/sinx`

Put the left hand side on a common denominator.

`sin^2x/sinx + cos^2x/sinx = 1/sinx`

`(sin^2x + cos^2x)/sinx= 1/sinx`

`1/sinx = 1/sinx`

`LHS = RHS`

Identity proved!

Hopefully this helps!