Question

Root 3 cosec140° - sec140°=4
prove that

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Answer 1

Explanation:

We are to show that
`√(3) cosec140^(0) - sec140^(0) = 4\\`

Proof:

From trigonometry identity;

`cosec \theta = (1)/(sin\theta) \\sec\theta = (1)/(cos\theta)`

`√(3) cosec140^(0) - sec140^(0) \\= (√(3) )/(sin140) - (1)/(cos140) \\= (√(3)cos140-sin140 )/(sin140cos140) \\`

From trigonometry, 2sinAcosA = Sin2A

`= (√(3)cos140-sin140 )/(sin140cos140) \\\\= (√(3)cos140-sin140 )/(sin280/2)\\= (4(√(3)/2cos140-1/2sin140) )/(2sin280)\\\\= (4(√(3)/2cos140-1/2sin140) )/(sin280)\\since sin420 = √(3)/2 \ and \ cos420 = 1/2 \\ then\\(4(sin420cos140-cos420sin140) )/(sin280)`

Also note that sin(B-C) = sinBcosC - cosBsinC

sin420cos140 - cos420sin140 = sin(420-140)

The resulting equation becomes;

`(4(sin(420-140)) )/(sin280)`

=
`(4sin280)/(sin280)\\ = 4 \ Proved!`