if tanθ+sinθ=m and tanθ-sinθ=n then prove m²-n²=√(mn)
Home
/
if tanθ+sinθ=m and tanθ-sinθ=n then prove m²-n²=√(mn)
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
Appropriate Question :-
If
[tex]\rm :\longmapsto\:tan\theta + sin\theta = m \: \: and \: \: tan\theta - sin\theta = n[/tex]
Prove that,
[tex]\rm :\longmapsto\: {m}^{2} - {n}^{2} = 4 \sqrt{mn} [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\rm :\longmapsto\:tan\theta + sin\theta = m[/tex]
and
[tex]\rm :\longmapsto\:tan\theta - sin\theta = n[/tex]
Consider,
[tex]\rm :\longmapsto\: {m}^{2} - {n}^{2} [/tex]
[tex]\rm \: = \: \: (m + n)(m - n)[/tex]
On substituting the values of m and n, we get
[tex]\rm \: = \: \: (tan\theta + sin\theta + tan\theta - sin\theta )(tan\theta + sin\theta - tan\theta + sin\theta )[/tex]
[tex]\rm \: = \: \: (2tan\theta )(2sin\theta )[/tex]
[tex]\rm \: = \: \: 4tan\theta sin\theta [/tex]
[tex]\bf\implies \: {m}^{2} - {n}^{2} = 4tan\theta sin\theta - - - (1)[/tex]
Now, Consider,
[tex]\rm :\longmapsto\: \sqrt{mn} [/tex]
[tex]\rm \: = \: \: \sqrt{(tan\theta + sin\theta )(tan\theta - sin\theta )} [/tex]
[tex]\rm \: = \: \: \sqrt{ {tan}^{2}\theta - {sin}^{2}\theta } [/tex]
[tex]\rm \: = \: \: \sqrt{\dfrac{ {sin}^{2} \theta }{ {cos}^{2} \theta } - {sin}^{2} \theta } [/tex]
[tex]\boxed{ \rm{ \because \: tanx = \frac{sinx}{cosx}}}[/tex]
[tex]\rm \: = \: \: \sqrt{ {sin}^{2} \theta ( {sec}^{2} \theta - 1)} [/tex]
[tex]\rm \: = \: \: \sqrt{ {sin}^{2} \theta {tan}^{2} \theta } [/tex]
[tex]\boxed{ \rm{ \because \: {sec}^{2}x = 1 + {tan}^{2}x}}[/tex]
[tex]\rm \: = \: \: sin\theta tan\theta [/tex]
Hence,
[tex]\bf\implies \:4 \sqrt{mn} = 4sin\theta tan\theta - - - - (2)[/tex]
From equation (1) and (2), we concluded that
[tex]\bf :\longmapsto\: {m}^{2} - {n}^{2} = 4 \sqrt{mn} [/tex]
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