prove the following identity showing all steps: \frac{cos(x+30)-sin(x+60)}{sin(x)cos(x)} =-sec(x) please help fas

Question

prove the following identity showing all steps:
\frac{cos(x+30)-sin(x+60)}{sin(x)cos(x)} =-sec(x)

please help fast

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Tryphena 4 years 2021-08-25T10:17:15+00:00 1 Answers 13 views 0

Answers ( )

  1. diemthu
    0
    2021-08-25T10:18:20+00:00
    Reply

    Answer:

    See solution below

    Step-by-step explanation:

    Given the expression

    \frac{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

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

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