Video: Evaluating Trigonometric Expressions given Trigonometric Equations

Find the value of tanΒ² πœƒ + cotΒ² πœƒ given tan πœƒ + cot πœƒ = 17.

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Video Transcript

Find the value of tan squared πœƒ plus cot squared πœƒ given tan πœƒ plus cot πœƒ is equal to 17.

We’re told that tan πœƒ plus cot πœƒ is equal to 17. We need to manipulate this equation so we can work out the value of tan squared πœƒ plus cot squared πœƒ. It might seem a little counterintuitive. But we’re going to square both sides of this equation. A common mistake is to think that that gives you tan squared πœƒ plus cot squared πœƒ. In fact, we think of squaring a number as multiplying it by itself. So we can say that tan πœƒ plus cot πœƒ squared is equal to tan πœƒ plus cot πœƒ multiplied by tan πœƒ plus cot πœƒ.

Well then, distribute these parenthesis by multiplying the first term in the first bracket by the first term in the second bracket. Tan πœƒ multiplied by tan πœƒ is tan squared πœƒ. We multiply the outer two terms and we get tan πœƒ cot πœƒ. Multiplying the inner terms, once again we get tan πœƒ cot πœƒ. And this part simplifies to two tan πœƒ cot πœƒ. We then multiply the last term in each bracket. And we get cot squared πœƒ. And we do have an expression of sorts that does involve tan squared πœƒ and cot squared πœƒ.

But what do we actually mean by cot πœƒ? Cot πœƒ is the same as one over tan πœƒ. We can say then that two tan πœƒ cot πœƒ is the same as two tan πœƒ multiplied by one over tan πœƒ. These tan πœƒs cancel. And we can see that tan πœƒ plus cot πœƒ squared is the same as tan squared πœƒ plus two plus cot squared πœƒ.

Let’s put this back into our original equation. We see that tan squared πœƒ plus two plus cot squared πœƒ is equal to 17 squared. And in fact, 17 squared is equal to 289. Next, we’ll subtract two from both sides of this equation. That will leave us with tan squared plus cot squared on the left-hand side as required. 289 minus two is 287. So we can see that tan squared πœƒ plus cot squared πœƒ is equal to 287. And that’s exactly what we we were looking to calculate.

The value of tan squared πœƒ plus cot squared πœƒ when tan πœƒ plus cot πœƒ is equal to 17 is 287.

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