trigonometry - Find $sin(A)$ and $cos(A)$ given $cos^4(A) - sin^4(A) = frac{1}{2}$ and $A$ is located in the second quadrant.

Question: Find $\sin(A)$ and $\cos(A)$, given $\cos^4(A)-\sin^4(A)=\frac{1}{2}$ and $A$ is located in the second quadrant.

Using the fundamental trigonometric identity, I was able to find that:

•　$\cos^2(A) - \sin^2(A) = \frac{1}{2}$

and

•

$$\cos(A) \cdot \sin(A) = -\frac{1}{4}$$

However, I am unsure about how to find $\sin(A)$ and $\cos(A)$ individually after this.

Edit: I solved the problem through using the Fundamental Trignometric Identity with the difference of $\cos^2(A)$ and $\sin^2(A)$.

Answer

Hint

$$\left( \cos(A)+ \sin(A) \right)^2 = 1+2 \sin(A) \cos(A)=\frac{1}{2} \\ \left( \cos(A)- \sin(A) \right)^2 = 1-2 \sin(A) \cos(A)=\frac{3}{2}$$

Take the square roots, and pay attention to the quadrant and the fact that $\cos^4(A) >\sin^4(A)$ to decide is the terms are positive or negative.

Alternate simpler solution

$$2 \cos^2(A)= \left( \cos^2(A)+\sin^2(A)\right)+\left( \cos^2(A)-\sin^2(A)\right)=1+\frac{1}{2} \\ 2 \sin^2(A)= \left( \cos^2(A)+\sin^2(A)\right)-\left( \cos^2(A)-\sin^2(A)\right)=1-\frac{1}{2} \\$$