# Video: Evaluating Trigonometric Expressions given Trigonometric Equations

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

02:08

### Video Transcript

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

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

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

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

Letβs put this back into our original equation. We see that tan squared π plus two plus cot squared π is equal to 17 squared. And in fact, 17 squared is equal to 289. Next, weβ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 plus cot squared π is equal to 287. And thatβs exactly what we we were looking to calculate.

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