Video: Simplifying Trigonometric Expression Using Cofunction Identities

Simplify cos πœƒ csc (90Β° βˆ’ πœƒ) βˆ’ tan πœƒ tan (90Β° βˆ’ πœƒ).

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Video Transcript

Simplify cos πœƒ times csc 90 degrees minus πœƒ minus tan πœƒ times tan 90 degrees minus πœƒ.

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

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

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

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œƒ by one over cos πœƒ to get cos πœƒ times one over cos πœƒ minus tan πœƒ times cot πœƒ. And we can use the definition of cot πœƒ: cot πœƒ is equal to one over tan πœƒ.

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

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

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