Question

Prove the identity note that each statement will be sent where

Answer 1

To prove:

`(sin(x+(\pi)/(2)))/(sin(\pi-x))=cotx`

Note that:

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

Applying these rules to the given trigonometric expression

`(sin(x+(\pi)/(2)))/(sin(\pi-x))=(sinxcos(\pi)/(2)+cosxsin(\pi)/(2))/(sin\pi cosx-cos\pi sinx)`

Note that:

cos(π) = -1

cos(π/2) = 0

sin(π) = 0

sin(π/2) = 1

Substitute these values into the expression above

`\begin{gathered} (s\imaginaryI n(x+(\pi)/(2)))/(cos(\pi- x))=(s\imaginaryI nx(0)+cosx(1))/((0)cosx-(-1)sinx) \\ \\ \frac{s\mathrm{i}n(x+(\pi)/(2))}{cos(\pi-x)}=\frac{cosx}{s\mathrm{i}nx} \\ \\ \frac{s\mathrm{i}n(x+(\pi)/(2))}{cos(\pi-x)}=cotx(Proved) \end{gathered}`