How do you prove the following problem step by step and which 4 identities are used to prove?

2/(sqrt3 cosx+sinx)=sec(pi/6-x)

3 Answers
Apr 8, 2018

Two identities needed

Explanation:

Well let's manipulate the right side to match the left, and name identities as encountered:

2/(sqrt3cosx+sinx)= sec(pi/6-x)

Reciprocal identity: secx= 1/cosx
2/(sqrt3cosx+sinx)= 1/cos(pi/6-x)

Cosine difference identity: cos(x-y)=cosxcosy+sinxsiny
2/(sqrt3cosx+sinx)= 1/(cos(pi/6)cosx+sin(pi/6)sinx)

Simplify:
2/(sqrt3cosx+sinx)= 1/((sqrt3cosx)/2+sinx/2)

2/(sqrt3cosx+sinx)= 1/((sqrt3cosx+sinx)/2)

Reciprocate:
2/(sqrt3cosx+sinx)= 2/(sqrt3cosx+sinx)

Apr 8, 2018

Image reference...

Explanation:

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Apr 8, 2018

Relations used

1.cos(pi/6)=sqrt3/2

2.sin(pi/6)=1/2

3.cosAcosB+sinAsinB=cos(A-B)

4.1/costheta=sectheta

LHS=2/(sqrt3 cosx+sinx)

=(2/2)/(sqrt3/2 cosx+1/2sinx)

=1/(cos(pi/6) cosx+sin(pi/6)sinx)

=1/(cos(pi/6-x))

=sec(pi/6-x)