Question

$\underline{\underline{\rm{\dotplus\: Question\::-}}}$
$\purple\dashrightarrow$$\sf{If \: tanA + sinA = m \: and \: tanA-sinA=n,show \: that \: {m}^{2} - {n}^{2} = \sqrt{mn}}$
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Answer 1

Appropriate Question

$\sf \: If \: tanA + sinA = m \: and \: tanA-sinA=n \\ \sf \: show \: that \: {m}^{2} \: - \: {n}^{2} \: = \: 4 \: \sqrt{mn} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

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

Given that,

$\rm :\longmapsto\:m = tanA + sinA$

and

$\rm :\longmapsto\:n = tanA - sinA$

Now, Consider

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

On substituting the values of m and n, we get

$\rm \:  =  \:  \: {(tanA + sinA)}^{2} - {(tanA - sinA)}^{2}$

We know

$\boxed{ \bf{ \: {(x + y)}^{2} - {(x - y)}^{2} = 4xy}}$

So, using this identity, we get

$\rm \:  =  \:  \: 4 \: tanA \: sinA$

$\bf\implies \: {m}^{2} - {n}^{2} = 4 \: tanA \: sinA - - - (1)$

Now, Consider

$\rm :\longmapsto\:4 \sqrt{mn}$

On substituting the values of m and n, we get

$\rm \:  =  \:  \: 4 \sqrt{(tanA + sinA)(tanA - sinA)}$

We know,

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

So, using this identity we get,

$\rm \:  =  \:  \: 4 \sqrt{ {tan}^{2}A - {sin}^{2}A }$

$\rm \:  =  \:  \: 4 \sqrt{ \dfrac{ {sin}^{2} A}{ {cos}^{2} A} - {sin}^{2}A }$

$\rm \:  =  \:  \: 4 \sqrt{ \bigg(\dfrac{1}{ {cos}^{2} A} - 1\bigg){sin}^{2}A }$

We know,

$\boxed{ \bf{ \: \frac{1}{cosx} = sinx}}$

So, using this

$\rm \:  =  \:  \: 4 \sqrt{ \bigg( {sec}^{2}A - 1\bigg){sin}^{2}A }$

We know,

$\boxed{ \bf{ \: {sec}^{2}x - {tan}^{2}x = 1}}$

So, using this, we get

$\rm \:  =  \:  \: 4 \sqrt{ {tan}^{2} A \times {sin}^{2} A}$

$\rm \:  =  \:  \: 4tanA \: sinA$

$\bf\implies \:4 \sqrt{mn} = 4 \: tanA \: sinA - - - (2)$

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

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

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