Question

If (m^2+n^2) sinθ = 2mn, then prove that cosθ = m^2+n^2/m^2-n^2

Answer 1

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

$\rm :\longmapsto\: ({m}^{2} + {n}^{2})sin\theta = 2mn$

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

$\rm :\longmapsto\:cos\theta = \dfrac{ {m}^{2} - {n}^{2} }{ {m}^{2} + {n}^{2} }$

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

$\boxed{ \red{\bf{ {sin}^{2}x + {cos}^{2}x = 1}}}$

$\boxed{ \red{\bf{ {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}}}$

$\boxed{ \red{\bf{ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy}}}$

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

Given that,

$\rm :\longmapsto\: ({m}^{2} + {n}^{2})sin\theta = 2mn$

$\rm :\implies\:sin\theta = \dfrac{2mn}{ {m}^{2} + {n}^{2} }$

We know,

$\rm :\longmapsto\: {sin}^{2}\theta + {cos}^{2}\theta = 1$

$\rm :\longmapsto\: {cos}^{2}\theta = 1 - {sin}^{2}\theta$

On substituting the value, we get

$\rm :\longmapsto\: {cos}^{2}\theta = 1 - {\bigg(\dfrac{2mn}{ {m}^{2} + {n}^{2} } \bigg) }^{2}$

$\rm :\longmapsto\: {cos}^{2}\theta = 1 - \dfrac{4 {m}^{2} {n}^{2}}{ {m}^{4} + {n}^{4} + 2 {m}^{2} {n}^{2} }$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{ {m}^{4} + {n}^{4} + 2 {m}^{2} {n}^{2} - 4 {m}^{2} {n}^{2}}{ {m}^{4} + {n}^{4} + 2 {m}^{2} {n}^{2} }$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{ {m}^{4} + {n}^{4} - 2 {m}^{2} {n}^{2}}{ {m}^{4} + {n}^{4} + 2 {m}^{2} {n}^{2} }$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{ {( {m}^{2} - {n}^{2})}^{2} }{ {( {m}^{2} + {n}^{2}) }^{2} }$

$\bf\implies \:cos\theta = \dfrac{ {m}^{2} - {n}^{2}}{ {m}^{2} + {n}^{2}}$

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