You can use the definitions of first three trigonometric ratios to find out the correct boxes.
The correct boxes filled would be like this
[tex]\rm \dfrac{12}{25} = \cos C \times \cos D = \sin C \times \sin D\\\\\dfrac{3}{5} = \sin C\\\\\dfrac{16}{25} = \cos C \times \tan D\\\\\dfrac{4}{5} = \cos C = \sin D[/tex]
What are the first three trigonometric ratios?
They are sine (sin), cosine (cos), tangent (tan)
They are ratios of sides of a right angled triangle, viewed from a fixed angle (not 90 degrees but the rest of the two angles)
Suppose we have the right angled triangle (as in given figure) CBD when angle B is right angle(of 90 degrees), then let the angle we choose be angle C.
Then we have:
The hypotenuse is always same, the slant line = CD
The side which is opposite to angle C is the line BD, called perpendicular from the viewpoint of angle C
And the line which is remaining now is called the base = CB
Thus, for these lines, we have the ratios as
[tex]sin(\angle C) = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}} = \dfrac{|BD|}{|CD|} = \dfrac{3}{5}\\\\cos(\angle C) = \dfrac{\text{Base}}{\text{Hypotenuse}} = \dfrac{|CB|}{|CD|} = \dfrac{4}{5}\\\\\\tan(\angle C) = \dfrac{\text{Perpendicular}}{\text{Base}} = \dfrac{|BD|}{|CB|} = \dfrac{3}{4}\\[/tex]
Using the above facts, we get the ratios as
From angle C:
[tex]sin(\angle C) = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}} = \dfrac{|BD|}{|CD|} = \dfrac{3}{5}\\\\cos(\angle C) = \dfrac{\text{Base}}{\text{Hypotenuse}} = \dfrac{|CB|}{|CD|} = \dfrac{4}{5}\\\\\\tan(\angle C) = \dfrac{\text{Perpendicular}}{\text{Base}} = \dfrac{|BD|}{|CB|} = \dfrac{3}{4}\\[/tex]
From angle D:
[tex]sin(\angle D) = \dfrac{\text{Perpendicular}}{\text{Hypotenuse}} = \dfrac{|CB|}{|CD|} = \dfrac{4}{5}\\\\cos(\angle D) = \dfrac{\text{Base}}{\text{Hypotenuse}} = \dfrac{|BD|}{|CD|} = \dfrac{3}{5}\\\\\\tan(\angle D) = \dfrac{\text{Perpendicular}}{\text{Base}} = \dfrac{|CB|}{|BD|} = \dfrac{4}{3}\\[/tex]
Thus,
[tex]sinC \times cosD = 3/5 \times 3/5 = 9/25[/tex]
[tex]tanC \times tanD = 4/3 \times 3/4 = 1[/tex]
[tex]cosC \times tanD = 4/5 \times 4/3 = 16/15[/tex]
Thus,
[tex]\rm \dfrac{12}{25} = \cos C \times \cos D = \sin C \times \sin D\\\\\dfrac{3}{5} = \sin C\\\\\dfrac{16}{25} = \cos C \times \tan D\\\\\dfrac{4}{5} = \cos C = \sin D[/tex]
Learn more about trigonometric ratios here;
https://brainly.com/question/22599614
