Prove that 1/sin(2theta) - cos(2theta)/sin(2theta) = tan(theta)

Before we begin to approach this question, we first write down all the relevant formulae for this trigonometry question. We can see that we will need the double angle formulae (on account of the presence of sin(2theta) and cos(2theta) so we will write them out.

sin(2theta) = 2sin(theta)*cos(theta)

cos(2theta) = cos²(2theta) - sin²(2theta)

Some other useful formulae are the cos²(theta) + sin²(theta) = 1 and sin(theta)/cos(theta) = tan(theta). These identities are almost always used in trigonometry questions, so it's well worth writing them down in case they need to be used.

In answering these trigonometric identities question, the accepted technique is to take the expression on the left hand side and transform it into the one on the right. We'll do this in multiple steps.

Step 1: We first factorise out 1/sin(2theta) so the equation reads (1/sin(2theta)(1-cos(2theta)) = tan(theta).
Step 2: Applying the double angle formulae for cosine and sine, the left hand side can then be expressed as (1/(2sin(theta)cos(theta)))(1-(cos²(theta) - sin²(theta))).
Step 3: Then, by applying our third equation, which equals to one, we can further transform and simplify the expression to (1/(2sin(theta)cos(theta)))*(2sin²(theta)).
Step 4: We can then cancel 2sin(theta) from both the numerator and the denominator in this expression which will simplify to sin(theta)/cos(theta). Step 5: Using the common identity above, we now know that the left hand side is equal to tan(theta), and so we have proven that the left hand side of the equation is equivalent to the right hand side.