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

Question

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

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Answer 1 · 2018-04-08T04:25:29

Two identities needed

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)#

Answer 2 · 2018-04-08T04:51:41

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Answer 3 · 2018-04-08T08:30:52

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)#