Question

sin⁴θ – cos⁴θ = 1 – 2cos²θ

Answer 1

$$\begin{gathered} {\sin }^4\theta -{\cos }^4\theta \\ ={\left({\sin }^2\theta \right)}^2-{\left({\cos }^2\theta \right)}^2 \\ =\left({\sin }^2\theta +{\cos }^2\theta \right)\left({\sin }^2\theta -{\cos }^2\theta \right)\left[a^2-b^2=\left(a-b\right)\left(a+b\right)\right] \end{gathered}$$
$$\begin{gathered} =1\times \left(1-{\cos }^2\theta -{\cos }^2\theta \right)\left({\sin }^2\theta +{\cos }^2\theta =1\right) \\ =1-2{\cos }^2\theta \end{gathered}$$