Problem 8: Prove that cos^2x = (cos x*cosec x)/(cotx + tan x) (4)

Expert Answers
marizi eNotes educator| Certified Educator

Note that cosec(x) can be express by csc(x)

The trigonometry identity: 

 Substitute` csc(x) =(1/sin(x))`   



Substitute `cot(x)= (cos(x)/sin(x))`

Then let `tan(x) =(1/cot(x))`

`= cot(x)/(cot(x)+(1/cot(x)))`


Substitute `1+cot^2(x) =csc(x)`


Take the reciprocal of the divisor (bottom) and proceed to multiplication



Substitute  `cot^2(x) = (cos^2(x))/(sin^2(x)) `  and 

`csc^2(x) = 1/(sin^2(x))`


Cancel out `sin^2(x)`

=`(cos^2(x))/1 or cos^2(x)`

This proves `cos^2` = 

justaguide eNotes educator| Certified Educator

The trigonometric identity `cos^2x = (cos x*cosec x)/(cotx + tan x)` has to be proved.

Start with the right hand side.

`(cos x*cosec x)/(cotx + tan x)`

Substitute `cot x = cos x/sin x` , `tan x = sin x/cos x` and `cosec x = 1/sin x`

= `(cos x*(1/sin x))/(cos x/sin x + sin x/cos x)`

= `(cos x/sin x)/((cos^2 x + sin^2x)/(sin x*cos x))`

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

Use the property `sin^2 + cos^2 = 1` .

= `cos^2x`

This proves that `cos^2x = (cos x*cosec x)/(cotx + tan x)`

marizi eNotes educator| Certified Educator

Take note the last part is:

This proves that `cos^2(x) = (cos(x)csc(x))/(cot(x)+tan(x))`