Video: Differentiating Trigonometric Functions Using Pythagorean Identities

If 𝑦 = 5(cosΒ² 2π‘₯ + sinΒ² 2π‘₯), find d𝑦/dπ‘₯.

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Video Transcript

If 𝑦 is equal to five multiplied by cos squared two π‘₯ plus sin squared two π‘₯, find d𝑦 by dπ‘₯.

In order to work out an expression for d𝑦 by dπ‘₯, we need to differentiate our equation with respect to π‘₯. At first glance, this appears quite complicated as we need to differentiate cos squared two π‘₯ and also sin squared two π‘₯. However, one of our trigonometric identities states that sin squared π‘₯ plus cos squared π‘₯ is equal to one. This also means that sin squared two π‘₯ plus cos squared two π‘₯ is also equal to one.

Our equation can be simplified to 𝑦 is equal to five multiplied by one. Five multiplied by one is equal to five. Therefore, 𝑦 equals five. We know that differentiating any constant gives us zero. This means that d𝑦 by dπ‘₯, in this case, equals zero. If 𝑦 is equal to five multiplied by cos squared two π‘₯ plus sin squared two π‘₯, then d𝑦 by dπ‘₯ equals zero.

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