Question

guys please help me in solving this question sin theta[1+tan theta] + cos theta [1+ cot theta]=sec theta + cosec theta​ please give answer only​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\bf \: Prove \: that \: \sf \: sin\theta \:(1 + tan\theta \:) + cos\theta \:(1 + cot\theta \:) = sec\theta \: + cosec\theta \:$

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

$\boxed{ \bf{ \: tan\theta \: = \dfrac{sin\theta \:}{cos\theta \:} }}$

$\boxed{ \bf{ \: cot\theta \: = \dfrac{cos\theta \:}{sin\theta \:} }}$

$\boxed{ \bf{ \: \dfrac{1}{cos\theta \:} = sec\theta \:}}$

$\boxed{ \bf{ \: \dfrac{1}{sin\theta \:} = cosec\theta \:}}$

$\boxed{ \bf{ \: {sin}^{2} \theta \: + {cos}^{2} \theta \: = 1}}$

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

Consider,

$\rm :\longmapsto\:\: sin\theta \:(1 + tan\theta \:) + cos\theta \:(1 + cot\theta \:)$

$\rm \: = \: \: sin\theta \: \bigg(1 + \dfrac{sin\theta \:}{cos\theta \:} \: \bigg) + cos\theta \: \bigg(1 + \dfrac{cos\theta \:}{sin\theta \:} \: \bigg)$

$\rm \: = \: \: sin\theta \: \bigg( \dfrac{cos\theta \: + sin\theta \:}{cos\theta \:} \: \bigg) + cos\theta \: \bigg(\dfrac{sin\theta \: + cos\theta \:}{sin\theta \:} \: \bigg)$

$\rm \: = \: (sin\theta \: + cos\theta \:)\bigg(\dfrac{sin\theta \:}{cos\theta \:} + \dfrac{cos\theta \:}{sin\theta \:} \bigg)$

$\rm \: = \: (sin\theta \: + cos\theta \:)\bigg( \dfrac{ {sin}^{2}\theta \: + {cos}^{2}\theta \: }{sin\theta \:cos\theta \:} \bigg)$

$\rm \: = \: (sin\theta \: + cos\theta \:) \times \dfrac{1}{sin\theta \:cos\theta \:}$

$\rm \: = \: \dfrac{ \cancel{sin\theta \:}}{ \cancel{sin\theta } \: \: cos\theta \:} + \dfrac{ \cancel{cos\theta \:}}{sin\theta \: \cancel{cos\theta \:}}$

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

$\rm \: = \: sec\theta \: + cosec\theta \:$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

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

Answer 2

Answer:

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Step-by-step explanation:

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