Answered

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Prove the identity note that each statement will be sent where

Prove The Identity Note That Each Statement Will Be Sent Where class=

Sagot :

To prove:

[tex]\frac{sin(x+\frac{\pi}{2})}{sin(\pi-x)}=cotx[/tex]

Note that:

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

Applying these rules to the given trigonometric expression

[tex]\frac{sin(x+\frac{\pi}{2})}{sin(\pi-x)}=\frac{sinxcos\frac{\pi}{2}+cosxsin\frac{\pi}{2}}{sin\pi cosx-cos\pi sinx}[/tex]

Note that:

cos(π) = -1

cos(π/2) = 0

sin(π) = 0

sin(π/2) = 1

Substitute these values into the expression above

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

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