Which double-angle or half-angle identity would you use to verify that 1+cos2a=2/1+tan2a?

In this case or scenario, the double-angle identity that should be used is the one for cosine. 

In totality, we shall need the following three trigonometric identities to end up with the equality: 

1. cos (2a) = cos² (a) - sin² (a)
2. sin² (a) + cos² (a) = 1 
3.  tan² (a) + 1 = sec² (a)

Using identities 1 and 2 on the left-hand side of the equation, we get the following:

1 + cos (2a) = 1 + cos² (a) - sin² (a) = 2 cos² (a)

Recalling that cos² (a) = 1 / sec² (a) and applying identity 3, we find the following:

2 cos² (a) = 2 / sec² (a) = 2 / (1 + tan² (a))

Therefore giving us:

2 cos² (a) = 2 / (1 + tan² (a))

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