Question

if tanθ+sinθ=m and tanθ-sinθ=n then prove m²-n²=√(mn)​

Answer 1

Appropriate Question :-

If

$\rm :\longmapsto\:tan\theta + sin\theta = m \: \: and \: \: tan\theta - sin\theta = n$

Prove that,

$\rm :\longmapsto\: {m}^{2} - {n}^{2} = 4 \sqrt{mn}$

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

Given that,

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

and

$\rm :\longmapsto\:tan\theta - sin\theta = n$

Consider,

$\rm :\longmapsto\: {m}^{2} - {n}^{2}$

$\rm \: = \: \: (m + n)(m - n)$

On substituting the values of m and n, we get

$\rm \: = \: \: (tan\theta + sin\theta + tan\theta - sin\theta )(tan\theta + sin\theta - tan\theta + sin\theta )$

$\rm \: = \: \: (2tan\theta )(2sin\theta )$

$\rm \: = \: \: 4tan\theta sin\theta$

$\bf\implies \: {m}^{2} - {n}^{2} = 4tan\theta sin\theta - - - (1)$

Now, Consider,

$\rm :\longmapsto\: \sqrt{mn}$

$\rm \: = \: \: \sqrt{(tan\theta + sin\theta )(tan\theta - sin\theta )}$

$\rm \: = \: \: \sqrt{ {tan}^{2}\theta - {sin}^{2}\theta }$

$\rm \: = \: \: \sqrt{\dfrac{ {sin}^{2} \theta }{ {cos}^{2} \theta } - {sin}^{2} \theta }$

$\boxed{ \rm{ \because \: tanx = \frac{sinx}{cosx}}}$

$\rm \: = \: \: \sqrt{ {sin}^{2} \theta ( {sec}^{2} \theta - 1)}$

$\rm \: = \: \: \sqrt{ {sin}^{2} \theta {tan}^{2} \theta }$

$\boxed{ \rm{ \because \: {sec}^{2}x = 1 + {tan}^{2}x}}$

$\rm \: = \: \: sin\theta tan\theta$

Hence,

$\bf\implies \:4 \sqrt{mn} = 4sin\theta tan\theta - - - - (2)$

From equation (1) and (2), we concluded that

$\bf :\longmapsto\: {m}^{2} - {n}^{2} = 4 \sqrt{mn}$

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