Verify each of the following by using equations (11.4), (12.2), and (12.3).

${\mathbf{cos}}^{\mathbf{4}}\mathbf{}\mathbf{z}\mathbf{+}{\mathbf{sin}}^{\mathbf{4}}\mathbf{z}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{sin}}^{\mathbf{2}}\mathbf{2}\mathbf{z}$

Given equation is ${\cos }^4\mathrm{z}+{\sin }^4\mathrm{z}=1-\frac{1}{2}{\sin }^{2}2\mathrm{z}$.

The term "Hyperbolic Function" refers to the relationship between a point on a hyperbola's distance from its origin and its coordinate axes, expressed as a function of an angle.

Given the equation is${\cos }^4\mathrm{z}+{\sin }^4\mathrm{z}=1-\frac{1}{2}{\sin }^{2}2\mathrm{z}$.

Takeleft hand side of the given equation${\cos }^4\mathrm{z}+{\sin }^4\mathrm{z}$and prove the right hand side.

Now, add and subtract$2{\cos }^2\mathrm{z}+{\sin }^{2}z$to the left hand side of the given equation.

$$\begin{gathered} ={\cos }^4\left(z\right)+{\sin }^4\left(z\right)+2{\cos }^2\left(z\right){\sin }^2\left(z\right)-2{\cos }^2\left(z\right){\sin }^2\left(z\right) \\ =\left[{\cos }^4\left(z\right)+{\sin }^4\left(z\right)+2{\cos }^2\left(z\right){\sin }^2\left(z\right)\right]-2{\cos }^2\left(z\right){\sin }^2\left(z\right) \\ =\left[{\cos }^4\left(z\right)+{\sin }^4\left(z\right)+2{\cos }^2\left(z\right){\sin }^2\left(z\right)\right]-2{\cos }^2\left(z\right){\sin }^2\left(z\right) \\ ={\left[{\cos }^2\left(z\right)+{\sin }^2\left(z\right)\right]}^2-2{\cos }^2\left(z\right){\sin }^2\left(z\right) \end{gathered}$$

Use property $co{s}^2\theta +si{n}^2\theta =1$in the above step.

$$\begin{gathered} =1-2{\cos }^2\left(z\right){\sin }^2\left(z\right) \\ =1-\frac{2}{2}\times {\cos }^2\left(z\right){\sin }^2\left(z\right) \\ =1-\frac{1}{2}\times 4{\cos }^2\left(z\right){\sin }^2\left(z\right) \\ =1-\frac{1}{2}{\left[2\cos \left(z\right)\sin \left(z\right)\right]}^2 \end{gathered}$$

Use property$2\cos \theta \sin \theta =\sin 2\theta$ in the above step.

$$\begin{gathered} =1-\frac{1}{2}{\left[\sin 2z\right]}^2 \\ =1-\frac{1}{2}{\sin }^2\left(2z\right) \end{gathered}$$

The result is equal to right hand side. Hence, the equation is verified.