It is given that 3cot A = 4
Or, cot A =\(\frac{4}{3}\)
Consider a right triangle ABC, right-angled at point B.
cot(A) = \(\frac{AB}{BC} =\frac{ 4}{3}\)
If AB is 4k, then BC will be 3k, where k is a positive integer.
In \(Δ\)ABC,
(AC)2 = (AB)2 + (BC)2
= (4k)2 + (3k)2
= 16k2 + 9k2
= 25k2
\(⇒\) AC = 5k
tan(A) = \(\frac{BC}{AB} =\frac{ 3}{4}\)
sin (A) = \(\frac{BC}{AC} = \frac{3}{5}\)
cos (A) = \(\frac{AB}{AC} =\frac{ 4}{5}\)
\(\frac{1-tan^2A}{1+tan^2A}=\frac{1-(\frac{3}{4})^2}{1+(\frac{3}{4})^2}\)
\(=\frac{1-\frac{9}{16}}{1+\frac{9}{16}}\)
\(=\frac{7}{25}\)
cos2 A - sin2 A \(=(\frac{4}{5})^2-(\frac{3}{5})^2\)
\(=\frac{16}{25}-\frac{9}{25}\)
\(=\frac{7}{25}\)
\(∴ \frac{(1-\text{tan}^2 A)}{(1+\text{tan}^2 A)} = \text{cos}^2 A –\text{ sin }^2 A\)
The relationship between the sides and angles of a right-angle triangle is described by trigonometry functions, sometimes known as circular functions. These trigonometric functions derive the relationship between the angles and sides of a triangle. In trigonometry, there are three primary functions of sine (sin), cosine (cos), tangent (tan). The other three main functions can be derived from the primary functions as cotangent (cot), secant (sec), and cosecant (cosec).
sin x = a/h
cos x = b/h
tan x = a/b
Tan x can also be represented as sin x/cos x
sec x = 1/cosx = h/b
cosec x = 1/sinx = h/a
cot x = 1/tan x = b/a