Question

If sin⁴ A – cos⁴ A = 1 and 0 < A ≤ 90°, then A = __________.

Answer 1

$$\begin{gathered} {\sin }^{4}A-{\cos }^{4}A=1 \\ \Rightarrow \left({\sin }^{2}A-{\cos }^{2}A\right)\left({\sin }^{2}A+{\cos }^{2}A\right)=1 \\ \Rightarrow {\sin }^{2}A-{\cos }^{2}A=1\left({\sin }^2\theta +{\cos }^2\theta =1\right) \\ \Rightarrow 1-{\sin }^{2}A+{\cos }^{2}A=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow 2{\cos }^{2}A=0 \\ \Rightarrow \cos A=0 \\ \Rightarrow A=90° \end{gathered}$$

If sin⁴ A – cos⁴ A = 1 and 0 < A ≤ 90°, then A = $\underline{90°}$.