Simplify sin 𝜃 cos 𝜃 over tan 𝜃 plus tan 𝜃 over sec 𝜃 cosec 𝜃.
The first thing to do is to rewrite this expression using sin 𝜃 and cos 𝜃 alone. We use the fact that tan 𝜃 is sin 𝜃 over cos 𝜃, that sec 𝜃 is one over cos 𝜃, and that cosec 𝜃 is one over sin 𝜃. Using these identities, we get the expression written only in terms of sin 𝜃 and cos 𝜃.
The next thing to do is to simplify. We have fractions in our fractions, and we’d really prefer not to. We multiply both numerator and denominator of the first fraction by cos 𝜃. The cos 𝜃s in the denominator cancel, and so we’re left with just sin 𝜃 in the denominator; we no longer have a fraction there.
In the second fraction, we multiply both numerator and denominator by cos 𝜃 times sin 𝜃. Of course, this doesn’t change the value of the fraction; it mainly allows us to simplify it. The cos 𝜃s in the denominator cancel out, as do the sin 𝜃s, and so we’re left with just a denominator of one times one, which is one. And the cos 𝜃s in the numerator also cancel.
Now we can tidy up. In the first fraction, the denominator is just sin 𝜃 after the cancellations, and we can simplify the numerator too, turning the cos 𝜃 times cos 𝜃 into a cos squared 𝜃. And in the second fraction, after the cancellation in the numerator, we get just sin 𝜃 times sin 𝜃, and the cancellations in the denominator gives us just one.
We can see in the first fraction that the factor of sin 𝜃 in the numerator cancels with the denominator, and so we’re left with just cos squared 𝜃. Doing the same for the second fraction, sin 𝜃 times sin 𝜃 over one is just sin 𝜃 times sin 𝜃 or sin squared 𝜃. And having simplified, hopefully we can recognize this expression: cos squared 𝜃 plus sin squared 𝜃 or, swapping the terms, sin squared 𝜃 plus cos squared 𝜃 is equal to one.
This is a famous identity, which is definitely worth remembering. Our final answer is therefore one. Sin 𝜃 times cos 𝜃 over tan 𝜃 plus tan 𝜃 over sec 𝜃 times cosec 𝜃 is equal to one for all values of 𝜃. We used these standards technique of simplifying a trigonometric expression like this. We rewrote everything in terms of sin 𝜃 and cos 𝜃.
Although we didn’t need it in this question, we should say for completeness that cot 𝜃 is equal to cos 𝜃 over sin 𝜃. Having then written this expression in terms of sin 𝜃 and cos 𝜃 alone, we found that we could cancel lots of factors. And having cancelled as far as we could, we then applied the identity sin squared 𝜃 plus cos squared 𝜃 is equal to one.