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How do you verify the identity #(1 + tan2u)(1 - sin2u) = 1#?

Answer 1

Identity does not exist.

Let #t=tan u# then using half angle formulae #tan 2u=(2t)/(1-t^2)#, #sin 2u= (2t)/(1+t^2)# and #cos 2u = (1-t^2)/(1+t^2)# The left hand side becomes #(1+(2t)/(1-t^2))(1-(2t)/(1+t^2))# this becomes #((1-t^2+2t)(1+t^2-2t))/((1-t^2)(1+t^2))# which is #((1-t^2+2t)(1-t)^2)/((1-t^2)(1+t^2))# which is #((1-t^2+2t))/((1+t^2))# which means that there is no identity

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Answer 2

Emery Adkins

To verify the identity (1 + tan^2(u))(1 - sin^2(u)) = 1, you can use trigonometric identities. Start with the expression (1 + tan^2(u))(1 - sin^2(u)). Expand it using the identity tan^2(u) = sec^2(u) - 1 and sin^2(u) + cos^2(u) = 1. After simplification, you should arrive at the expression 1, proving the identity.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.