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Question Video: Using the Pythagorean Identities to Evaluate a Trigonometric Function for an Angle Mathematics • 10th Grade

Find the value of secยฒ ๐œƒ given tanยฒ ๐œƒ = 4.

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Video Transcript

Find the value of sec squared ๐œƒ given tan squared ๐œƒ equals four.

To answer this question, weโ€™re not going to solve the equation tan squared ๐œƒ equals four for ๐œƒ. Instead, weโ€™re going to find a Pythagorean identity that links sec squared ๐œƒ and tan squared ๐œƒ.

To find this identity, weโ€™re going to recall one of the most straightforward trigonometric identities. That is, sin squared ๐œƒ plus cos squared ๐œƒ equals one. We also know that tan ๐œƒ is equal to sin ๐œƒ over cos ๐œƒ. So it follows that tan squared ๐œƒ must also be equal to sin squared ๐œƒ over cos squared ๐œƒ.

And so what weโ€™re going to do is individually divide each of the elements in our first identity by cos squared ๐œƒ. When we do, we can write sin squared ๐œƒ divided by cos squared ๐œƒ is tan squared ๐œƒ. Cos squared ๐œƒ divided by cos squared ๐œƒ is one. And since one over cos ๐œƒ is sec ๐œƒ, we know one over cos squared ๐œƒ is sec squared ๐œƒ.

And so this is the Pythagorean identity weโ€™re going to use. And in fact, we donโ€™t need to derive it each time. We can simply quote that sec squared ๐œƒ is equal to tan squared ๐œƒ plus one. In this question, tan squared ๐œƒ is equal to four. So we can say that sec squared ๐œƒ must be equal to four plus one, which is simply five. And so the value of sec squared ๐œƒ given that tan squared ๐œƒ is equal to four is five.

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