  ### Home > APCALC > Chapter 5 > Lesson 5.3.3 > Problem5-135

5-135.

If $y = (\operatorname{cos}(x) + \operatorname{sin}(x))^2$, find $\frac { d y } { d x }$ using two different methods. Then demonstrate that the two results are equal.

Common Trigonometric Identities

 Reciprocal $\sec(θ)=\frac{1}{\cos(\theta)}$ $\csc(θ)=\frac{1}{\sin(\theta)}$$\tan(θ)=\frac{\sin(\theta)}{\cos(\theta)}$ $\cot(θ)=\frac{\cos(\theta)}{\sin(\theta)}$ Pythagorean$\sin^2\left(θ\right)+\cos^2\left(θ\right)=1$$\tan^2\left(θ\right)+1=\sec^2\left(θ\right)$$1+\cot^2\left(θ\right)=\csc^2\left(θ\right)$ Angle Sum$\sin\left(a\pm b\right)=\sin\left(a\right)\cos\left(b\right)\pm\cos\left(a\right)\sin\left(b\right)$$\cos\left(a\pm b\right)=\cos\left(a\right)\cos\left(b\right)∓\sin\left(a\right)\sin\left(b\right)$ Double Angle$\sin\left(2a\right)=2\sin\left(a\right)\cos\left(a\right)$$\operatorname { cos } ( 2 a ) = \left\{ \begin{array} { l } { \operatorname { cos } ^ { 2 } ( a ) - \operatorname { sin } ^ { 2 } ( a ) } \\ { 2 \operatorname { cos } ^ { 2 } ( a ) - 1 } \\ { 1 - 2 \operatorname { sin } ^ { 2 } ( a ) } \end{array} \right.$

Method 1: Before you differentiate, expand and simplify using trig identities. Then use the Product Rule to differentiate.
Method 2: Chain Rule.

The results might not look the same. This is where you get to practice proving a trig identity (in other words, proving that the two results are equivalent.)