Question

Prove the identity using the following rules, Algebra, reciprocal, quotient, Pythagorean, and odd/even?

`csc(-x)/sec(-x)+cos(-x)/sin(-x)=-2cotx`

Answer 1

check the explanation below

Explanation:

`csc(-x)/sec(-x)=(1/sin(-x))/(1/cos(-x))=cos(-x)/sin(-x)` `color(green)(rarr "reciprocal"`

now substitute

`cos(-x)/sin(-x)+cos(-x)/sin(-x)=2cos(-x)/sin(-x)`

`color(blue)(sinx " is odd so "sin(-x)=-sinx"`

`color(blue)( cosx " is even so "cos(-x)=cosx`

Substitute

`=2cosx/-sinx` `color(green)(rarreven and odd " functions"`

`=-2cot(x)` `color(green)(rarrquoitent`

I hope this was helpful.