Question

please solve the above question​

Answer 1

$\purple{\large\underline{\sf{Given \:Question - }}}$

Prove that

$\sf \: \dfrac{1 + 2sin\theta + {sin}^{2} \theta }{ {cos}^{2} \theta } = \dfrac{sec\theta + tan\theta }{sec\theta - tan\theta }$

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

Consider, RHS

$\rm :\longmapsto\:\dfrac{sec\theta + tan\theta }{sec\theta - tan\theta }$

On rationalizing the denominator, we get

$\rm \: = \: \dfrac{sec\theta + tan\theta }{sec\theta - tan\theta } \times \dfrac{sec\theta + tan\theta }{sec\theta + tan\theta }$

$\rm \: = \: \dfrac{ {(sec\theta + tan\theta )}^{2} }{ {sec}^{2} \theta - {tan}^{2} \theta }$

We know,

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

and

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

So, using these, Identities, we get

$\rm \: = \: \dfrac{ {sec}^{2} \theta + {tan}^{2} \theta + 2sec\theta tan\theta }{1}$

We know,

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

and

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

So, using these we get,

$\rm \: = \: {\bigg[\dfrac{1}{cos\theta }\bigg]}^{2} + {\bigg[\dfrac{sin\theta }{cos\theta }\bigg]}^{2} + 2 \times \dfrac{1}{cos\theta } \times \dfrac{sin\theta }{cos\theta }$

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

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

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

Hence,

$\boxed{ \bf{ \:\dfrac{1 + 2sin\theta + {sin}^{2} \theta }{ {cos}^{2} \theta } = \dfrac{sec\theta + tan\theta }{sec\theta - tan\theta } }}$

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