Prove that 1/(tanx) + tanx = 1/sinxcosx

The key here is to realise that tanx = sinx/cosx. If we write out the left hand side of the equation in terms of sine and cosine we get:

cosx/sinx + sinx/cosx

These two fractions can be put over a common denominator of sinxcosx to give:

(cos2x + sin2x)/sinxcosx

If we then use the well-known identity cos2x + sin2x = 1, we see that the above expression is equivalent to 1/sinxcosx, which is the expression we were required to find.

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