Answer:
OPTION C: Sin C - Cos C = s - r
Step-by-step explanation:
ABC is a right angled triangle. ∠A = 90°, from the figure.
Therefore, BC = hypotenuse, say h
Now, we find the length of AB and AC.
We know that: ![$ \textbf{Sin A} = \frac{\textbf{opp}}{\textbf{hyp}} $](https://tex.z-dn.net/?f=%24%20%5Ctextbf%7BSin%20A%7D%20%3D%20%20%5Cfrac%7B%5Ctextbf%7Bopp%7D%7D%7B%5Ctextbf%7Bhyp%7D%7D%20%24)
and ![$ \textbf{Cos A} = \frac{\textbf{adj}}{\textbf{hyp}} $](https://tex.z-dn.net/?f=%24%20%5Ctextbf%7BCos%20A%7D%20%3D%20%5Cfrac%7B%5Ctextbf%7Badj%7D%7D%7B%5Ctextbf%7Bhyp%7D%7D%20%24)
Given, Sin B = r and Cos B = s
⇒ ![$ Sin B = r = \frac{opp}{hyp} = \frac{AC}{BC} = \frac{AC}{h} $](https://tex.z-dn.net/?f=%24%20Sin%20B%20%3D%20r%20%3D%20%5Cfrac%7Bopp%7D%7Bhyp%7D%20%3D%20%5Cfrac%7BAC%7D%7BBC%7D%20%3D%20%5Cfrac%7BAC%7D%7Bh%7D%20%24)
⇒ ![$ \textbf{AC} = \textbf{rh} $](https://tex.z-dn.net/?f=%24%20%5Ctextbf%7BAC%7D%20%3D%20%5Ctextbf%7Brh%7D%20%24)
Hence, the length of the side AC = rh
Now, to compute the length of AB, we use Cos B.
![$ Cos B = s = \frac{adj}{hyp} = \frac{AB}{BC} = \frac{AB}{h} $](https://tex.z-dn.net/?f=%24%20Cos%20B%20%3D%20s%20%3D%20%5Cfrac%7Badj%7D%7Bhyp%7D%20%3D%20%5Cfrac%7BAB%7D%7BBC%7D%20%3D%20%5Cfrac%7BAB%7D%7Bh%7D%20%24)
⇒ ![$ \textbf{AB} = \textbf{sh} $](https://tex.z-dn.net/?f=%24%20%5Ctextbf%7BAB%7D%20%3D%20%5Ctextbf%7Bsh%7D%20%24)
Hence, the length of the side AB = sh
Now, we are asked to compute Sin C - Cos C.
![$ Sin C = \frac{opp}{hyp} $](https://tex.z-dn.net/?f=%24%20Sin%20C%20%3D%20%5Cfrac%7Bopp%7D%7Bhyp%7D%20%24)
⇒ ![$ Sin C = \frac{AB}{BC} $](https://tex.z-dn.net/?f=%24%20Sin%20C%20%3D%20%5Cfrac%7BAB%7D%7BBC%7D%20%24)
![$ = \frac{sh}{h} $](https://tex.z-dn.net/?f=%24%20%3D%20%5Cfrac%7Bsh%7D%7Bh%7D%20%24)
= s
Sin C = s
![$ Cos C = \frac{adj}{hyp} $](https://tex.z-dn.net/?f=%24%20%20Cos%20C%20%3D%20%5Cfrac%7Badj%7D%7Bhyp%7D%20%24)
![$ \implies Cos C = \frac{AC}{BC} $](https://tex.z-dn.net/?f=%24%20%5Cimplies%20Cos%20C%20%3D%20%5Cfrac%7BAC%7D%7BBC%7D%20%24)
⇒ Cos C = ![$ \frac{rh}{h} $](https://tex.z-dn.net/?f=%24%20%5Cfrac%7Brh%7D%7Bh%7D%20%24)
Therefore, Cos C = r
So, Sin C - Cos C = s - r, which is OPTION C and is the right answer.