Question

if secA+tanA=x then what is the value of cosecA?

Answer 1

$\large\underline{\sf{Given \:Question - }}$

$\: \: \: \:\sf \: secA + tanA = x$

$\large\underline{\sf{To\:Find - }}$

$\sf \: \: cosecA$

$\begin{gathered}\Large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

$1. \: \sf \: {sec}^{2} A - {tan}^{2} A = 1$

$2. \: \: \sf \: tanA = \: \dfrac{sinA}{cosA}$

$3. \: \: \sf \: secA = \dfrac{1}{cosA}$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:secA + tanA = x - - - (1)$

We know,

$\rm :\longmapsto\: {sec}^{2} A - {tan}^{2} A = 1$

$\rm :\longmapsto\:(secA + tanA)(secA - tanA) = 1$

$\rm :\longmapsto\:x \times (secA - tanA) = 1$

$\bf\implies \:secA - tanA = \dfrac{1}{x} - - - (2)$

Now, Adding equation (1) and (2), we get

$\rm :\longmapsto\:2secA = x + \dfrac{1}{x}$

$\bf\implies \:secA = \dfrac{ {x}^{2} + 1 }{2x} - - - (3)$

Now, Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\:2tanA = x - \dfrac{1}{x}$

$\bf\implies \:tanA = \dfrac{ {x}^{2} - 1 }{2x} - - - (4)$

Now,

Consider,

$\rm :\longmapsto\:sinA$

$\sf \: = \: sinA \times \dfrac{cosA}{cosA}$

$\sf \: = \: \dfrac{sinA}{cosA} \times cosA$

$\sf \: = \: tanA \times \dfrac{1}{secA}$

$\sf \: = \: \dfrac{ {x}^{2} - 1 }{\cancel{2x}} \times \dfrac{\cancel{2x}}{ {x}^{2} + 1 }$

$\sf \: = \: \dfrac{ {x}^{2} - 1}{ {x}^{2} + 1}$

Therefore,

$\rm :\longmapsto\:cosecA = \dfrac{1}{sinA} = \dfrac{ {x}^{2} + 1}{ {x}^{2} - 1}$

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