Question

if (1+4x^2)cosA=4x,then let us prove that, cos^2+cos^2=1.

Answer 1

cos^4x=(cos^2x)^2=(1-sin^2x)^2=1+ sin^4x-2sin^2x 

=>sin^4x / a +cos^4x / b=1/a+b 


=>sin^4x/a + (1+sin^4x-2sin^2x)/b = 1/(a+b) 

[b*sin^4x + a(sin^4x-2sin^2x+1)] /ab = 1/(a+b) 

=>[(a+b)sin^4x-2a sin^2x+a]/ab = 1/a+b 

=>(a+b)^2 sin^4x - 2a(a+b)sin^2x + a(a+b) =ab 

=>(a+b)^2 sin^4x - 2a(a+b)sin^2x + a^2 


=> [(a+b)sin^2x-a]^2 = 0 
=>(a+b)sin^2x - a = 0 

sin^2x=a/(a+b).........(1) 
(take fourth power of both side) 
=>sin^8x=a^4/(a+b)^4 
(divide by a^3 both side) 
=>sin^8x/a^3=a/(a+b)^4. 
........(2) 
=>cos^2x=1 - sin^2x=1-a/(a+b)=b/(a+b). (from eq 1 substituting value of sin^2x) 

=>cos^2x=b/(a+b)...........(3) 
(take fourth power of both side) 

=>cos^8x=b^4/(a+b)^4 
(divide by b^3 both side) 
=>cos^8x/b^3 =b/(a+b)^4......(4) 

(adding eq 2@4) 
=>sin^8x/a^3 + cos^8x/b^3=a/(a+b)^4 + b/(a+b)^4 =(a+b)/(a+b)^4=1/(a+b)^3 ........proved