If 2Cosθ = x+1/x So, Prove That ? Cos2θ = 1/2(x²+1/x²)

2 Answers
P dilip_k
May 20, 2018

#LHS=cos2theta#

#=2cos^2theta-1#

#=2(1/2(x+1/x))^2-1#

#=1/2(x^2+1/x^2+2*x*1/x)-1#

#=1/2(x^2+1/x^2)+1/2*2-1#

#=1/2(x^2+1/x^2)=RHS#

Sonnhard
May 20, 2018

Sqaring the given equation we get
#cos(theta)=1/4*(x^2+1/x^2+2)# then by multiplying by #2# we get
#2cos^2(theta)=1/2*(x^2+1/x^2)+1# from here we get
#2cos^2(theta)-1=1/2(x^2+1/x^2)#

Explanation:

we used #(x+1/x)^2=x^2+1/x^2+2#
#cos(2x)=2cos^2(x)-1#

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