Question

Simplify: (help me please)?

sin(360-x) x tan(90-x)

Answer 1

`=-cosx`

Explanation:

The sine function is periodic with period `360^@`, hence

`sin(360^@-x) =sin(-x)=-sinx`

About the tangent function; let `y=90^@ - x => x+y=90^@`.
`x` and `y` are complementary angles. We usually meet them in right triangles, where trigonometric functions are usually more familiar. As such, let `x` and `y` be the acute angles of a right triangle with sides `cc a, ccb, cc c`.

`tanx="opposite"_x/"adjacent"_x=ccb/cca`

`cotx="adjacent"_x/"opposite"_x=cca/ccb`

`tany=tan(90^@-x)="opposite"_y/"adjacent"_y = cca/ccb=cotx`

`:. tan(90^@-x)=cotx`

Hence,

`sin(360^@-x)xxtan(90^@-x)=-sinx xx cotx=-sinx xx cosx/sinx=-cosx`