Question

Eliminate θ between the equations: cos θ + sin θ = m and sec θ + cosec θ = n.​

Answer 1

Given Question :-

Eliminate θ between the equations:

• cos θ + sin θ = m and sec θ + cosec θ = n.

ANSWER

GIVEN :-

• cos θ + sin θ = m

• sec θ + cosec θ = n.

TO ELIMINATE :-

• Eliminate θ

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

$(1). \: \boxed{ \bf \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }$

$(2). \: \boxed{ \bf \: {sin}^{2} x + {cos}^{2} x = 1}$

$(3). \: \boxed{ \bf \: cosecx = \dfrac{1}{sinx} }$

$(4). \: \boxed{ \bf \:secx = \dfrac{1}{cosx} }$

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

Consider,

$\rm :\longmapsto\:cosec\theta \: + sec\theta \: = n$

$\rm :\longmapsto\:\dfrac{1}{sin\theta \:} + \dfrac{1}{cos\theta \:} = n$

$\rm :\longmapsto\:\dfrac{cos\theta \: + sin\theta \:}{sin\theta \:cos\theta \:} = n$

$\rm :\implies\:\dfrac{m}{sin\theta \:cos\theta \:} = n \: \: \: \: \: \: ( \because \: sin\theta \: + cos\theta \: = m)$

$\bf\implies \:sin\theta \:cos\theta \: = \dfrac{m}{n} - - - - (1)$

Now,

• It is given that

$\rm :\longmapsto\:sin\theta \: + cos\theta \: = m$

• On squaring both sides, we get

$\rm :\longmapsto\: {sin}^{2} \theta \: + {cos}^{2} \theta \: + 2sin\theta \:cos\theta \: = {m}^{2}$

$\rm :\implies\:1 + 2 \times \dfrac{m}{n} = {m}^{2} \: \: \: \: \: \: \: \: \: \: \: ( {\because} \: of \: (1))$

$\bf\implies \:2m = n( {m}^{2} - 1)$

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