Step-by-step explanation:
[tex]\large\underline{\sf{Solution-}}[/tex]
[tex]\rm \longmapsto\:3cot\theta = 4[/tex]
[tex]\rm\implies \:cot\theta = \dfrac{4}{3} [/tex]
Now, Consider
[tex]\rm \longmapsto\:\dfrac{3sin\theta + 4cos\theta }{3sin\theta - 4cos\theta } [/tex]
[tex]\rm \: = \: \dfrac{sin\theta \bigg[3 + 4\dfrac{cos\theta }{sin\theta } \bigg]}{sin\theta \bigg[3 - 4\dfrac{cos\theta }{sin\theta } \bigg]} [/tex]
[tex]\rm \: = \: \dfrac{3 + 4cot\theta }{3 - 4cot\theta } [/tex]
[tex]\rm \: = \: \dfrac{3 + 4 \times \dfrac{4}{3} }{3 - 4 \times \dfrac{4}{3} } [/tex]
[tex]\rm \: = \: \dfrac{3 + \dfrac{16}{3} }{3 - \dfrac{16}{3} } [/tex]
[tex]\rm \: = \: \dfrac{\dfrac{9 + 16}{3} }{ \dfrac{9 - 16}{3} } [/tex]
[tex]\rm \: = \: \dfrac{25}{ - 7} [/tex]
[tex]\rm \: = \: - \: \dfrac{25}{7} [/tex]
Hence,
[tex]\rm \longmapsto\:\boxed{\tt{ \dfrac{3sin\theta + 4cos\theta }{3sin\theta - 4cos\theta } = - \frac{25}{7} \: }} \\ [/tex]
Additional Information:-
Relationship between sides and T ratios
sin θ = Opposite Side/Hypotenuse
cos θ = Adjacent Side/Hypotenuse
tan θ = Opposite Side/Adjacent Side
sec θ = Hypotenuse/Adjacent Side
cosec θ = Hypotenuse/Opposite Side
cot θ = Adjacent Side/Opposite Side
Reciprocal Identities
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
Co-function Identities
sin (90°−x) = cos x
cos (90°−x) = sin x
tan (90°−x) = cot x
cot (90°−x) = tan x
sec (90°−x) = cosec x
cosec (90°−x) = sec x
Fundamental Trigonometric Identities
sin²θ + cos²θ = 1
sec²θ - tan²θ = 1
cosec²θ - cot²θ = 1
