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Problem 55 Hard Difficulty

Differentiate each trigonometric identity to obtain a new (or familiar) identity.
(a) $ \tan x = \frac {\sin x}{\cos x} $
(b) $ \sec x = \frac {1}{\cos x} $
(c) $ \sin x + \cos x = \frac {1 + \cot x}{\csc x} $

Answer

a) $\sec ^{2} x=\frac{1}{\cos ^{2} x}$
b) $\tan x \sec x$
c) $\cos x-\sin x=\frac{\cot x-1}{\csc x}$

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Video Transcript

Yes, Claire. So when you right here. So we know that the derivative of tan seek it square. So the sequence square gives us co sign. Where is the derivative of sign? Minus sign times the derivative of co sign over. Co sign square. Always simplifying. Get co signed Square signed. Square over. Co sign square and this gives us one over co signed square report Be we're gonna difference She the derivative of secret is equal to seek it. Times tangent We're gonna apply the quotient room, get the co sign, comes the derivative of one my ass One times the derivative of co sign over co sign square. This becomes equal to sign Thanks over co sign square for part C, we have signed X. Let's go Sign is equal to one plus cu o tangent over CO c Good. We're gonna differentiate both sides in respect to X using the quotient rule get co sign of X minus Sign of acts is equal to seek it. Hi. Negative. Cosi can't square minus one plus go to engine times. Negative, but it go seek. It turns court engine all over. Go seek it square it says equal to co sign minus sign is equal to negative Co. C Can't Square Klesko 10 Gin plus coach Engine square over Co Seek it. Wait. Have to use our identity of co Tangent Square is equal to CO C can't square minus one. This becomes equal to we'll sign minus sign. It is equal to this. The cones CO C can't square minus one. So only simplify. I get coat engine My last one over Costa Rican.