Question

The value of $[{\cos }^4\mathrm{A}-{\sin }^4\mathrm{A}]$ is equal to

Answer 1

Determine the value of ${\cos }^4\mathrm{A}-{\sin }^4\mathrm{A}$.

Simplify the trigonometry identity as follow:

$$\begin{gathered} {\cos }^4\mathrm{A}-{\sin }^4\mathrm{A}={\left({\cos }^2\mathrm{A}\right)}^2-{\left({\sin }^2\mathrm{A}\right)}^2 \\ =\left({\cos }^2\mathrm{A}-{\sin }^2\mathrm{A}\right)\left({\cos }^2\mathrm{A}+{\sin }^2\mathrm{A}\right)...\left[\because a^2–b^2=(a+b)(a–b)\right] \\ =\left({\cos }^2\mathrm{A}-{\sin }^2\mathrm{A}\right).1...\left[\because \left(co{s}^{2}A+si{n}^{2}A\right)=1\right] \\ =\cos 2\mathrm{A}...\left[\because cos2A=co{s}^{2}A-si{n}^{2}A\right] \end{gathered}$$

Hence, the required value is ${\cos }^4\mathrm{A}-{\sin }^4\mathrm{A}=\cos 2\mathrm{A}$