# Video: Simplifying Trigonometric Expressions

Simplify sin π cos π/tan π + tan π/sec π csc π.

03:02

### Video Transcript

Simplify sin π cos π over tan π plus tan π over sec π cosec π.

The first thing to do is to rewrite this expression using sin π and cos π alone. We use the fact that tan π is sin π over cos π, that sec π is one over cos π, and that cosec π is one over sin π. Using these identities, we get the expression written only in terms of sin π and cos π.

The next thing to do is to simplify. We have fractions in our fractions, and weβd really prefer not to. We multiply both numerator and denominator of the first fraction by cos π. The cos πs in the denominator cancel, and so weβre left with just sin π in the denominator; we no longer have a fraction there.

In the second fraction, we multiply both numerator and denominator by cos π times sin π. Of course, this doesnβt change the value of the fraction; it mainly allows us to simplify it. The cos πs in the denominator cancel out, as do the sin πs, and so weβre left with just a denominator of one times one, which is one. And the cos πs in the numerator also cancel.

Now we can tidy up. In the first fraction, the denominator is just sin π after the cancellations, and we can simplify the numerator too, turning the cos π times cos π into a cos squared π. And in the second fraction, after the cancellation in the numerator, we get just sin π times sin π, and the cancellations in the denominator gives us just one.

We can see in the first fraction that the factor of sin π in the numerator cancels with the denominator, and so weβre left with just cos squared π. Doing the same for the second fraction, sin π times sin π over one is just sin π times sin π or sin squared π. And having simplified, hopefully we can recognize this expression: cos squared π plus sin squared π or, swapping the terms, sin squared π plus cos squared π is equal to one.

This is a famous identity, which is definitely worth remembering. Our final answer is therefore one. Sin π times cos π over tan π plus tan π over sec π times cosec π is equal to one for all values of π. We used these standards technique of simplifying a trigonometric expression like this. We rewrote everything in terms of sin π and cos π.

Although we didnβt need it in this question, we should say for completeness that cot π is equal to cos π over sin π. Having then written this expression in terms of sin π and cos π 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 π plus cos squared π is equal to one.