prove that cos(tan-1{cos(tan-1x)}) = (x^2+1/x^2+2)^1/2

Hi there.. Write tan-¹x as cos-¹1/√1+x².. the ques reduces to cos(tan-¹(1/√1+x²). Again write tan-¹1/√1+x² as cos-¹(√1+x²/√2+x²)... I am able to write this.. for simplification you can let tan-¹x as y and find value of cosy..