Question

Prove the following identity showing all steps:


`(cos(x+30)-sin(x+60))/(sin(x)cos(x)) =-sec(x)`

please help fast

Answer 1

See solution below

Explanation:

Given the expression

`(cos(x+30)-sin(x+60))/(sin(x)cos(x)) \\`

Recall that

cos x = sin(90-x)

cos(x+30 ) = sin (90-(x+30)

= sin(90-x-30)

= sin(60-x)

Substitute

`(sin(60-x)-sin(x+60))/(sin(x)cos(x)) \\= (sin60cosx-cos60sinx)-sinxcos60-cosxsin60))/(sin(x)cos(x)) \\= (-2cos60sinx))/(sin(x)cos(x)) \\= (-2(1/2)sinx))/(sin(x)cos(x)) \\= (-1)/(cos(x))\\= (1)/(cos(x))\\ \\= -sec(x) Proved`