Question Video: Simplifying Trigonometric Expression Using Cofunction Identities Mathematics • 10th Grade

Simplify cos π csc (90Β° β π) β tan π tan (90Β° β π).

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Video Transcript

Simplify cos π times csc 90 degrees minus π minus tan π times tan 90 degrees minus π.

Whenever you see something like csc 90 degrees minus π or tan 90 degrees minus π or cos 90 degrees minus π, some trigonometric function evaluated at 90 degrees minus π, you should think about cofunction identities. And here they all are.

This identity tells us that we can rewrite csc 90 degrees minus π as sec π. So making this substitution, we get cos π times sec π minus tan π times tan 90 degrees minus π.

And this identity tells us that we can replace tan 90 degrees minus π by cot π, like so. So now we have cos π times sec π minus tan π times cot π.

We can use the definition of the reciprocal trigonometric functions to write this expression in terms of sin, cos, and tan alone. For example, we can replace sec π by one over cos π to get cos π times one over cos π minus tan π times cot π. And we can use the definition of cot π: cot π is equal to one over tan π.

And now we have an expression, which is written in terms only of cos π and tan π. We can see that thereβs further simplification to be had here. Weβve got cos π times one over cos π, and so those cos thetas cancel out.

And similarly, in tan π times one over tan π, the tan thetas cancel out. This leaves us simply with one minus one. So the answer is zero; cos π times csc 90 degrees minus π minus tan π times tan 90 degrees minus π is equal to zero.