Use the given figure to find :

(i) tan θ°

(ii) θ°

(iii) sin² θ° - cos² θ°

(iv) Use sin θ° to find the value of x.

(i) tan θ°

tan θ° = $\dfrac{Perpendicular}{Base}$

= $\dfrac{AB}{BC} = \dfrac{5}{5} = 1$

Hence, tan θ° = 1.

(ii) θ°

⇒ tan θ° = tan 45°

So, θ° = 45°

Hence, θ° = 45°.

(iii) sin² θ° - cos² θ°

sin θ° = $\dfrac{Perpendicular}{Hypotenuse}$

= $\dfrac{AB}{AC} = \dfrac{5}{x}$

cos θ° = $\dfrac{Base}{Hypotenuse}$

= $\dfrac{BC}{AC} = \dfrac{5}{x}$

Now, sin² θ° - cos² θ°

$= \Big(\dfrac{5}{x}\Big)^2 - \Big(\dfrac{5}{x}\Big)^2\\[1em] = \dfrac{25}{x^2} - \dfrac{25}{x^2}\\[1em] = 0$

Hence, sin² θ° - cos² θ° = 0.

(iv) Use sin θ° to find the value of x.

sin θ° = $\dfrac{Perpendicular}{Hypotenuse} = \dfrac{AB}{AC} = \dfrac{5}{x}$

sin θ° = sin 45° = $\dfrac{1}{\sqrt2}$

So, $\dfrac{1}{\sqrt2} = \dfrac{5}{x}$

x = $5\sqrt2$ = 7.07

Hence, x = $5\sqrt2$ = 7.07.