Question

Prove that (1 + tan theta) ^ 2 + (1 + cot theta) ^ 2 = (sec(theta) + cos e * c * theta) ^ 2 ".​

Answer 1

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

$\bf \: Prove \: that \sf \: {(1 + tan\theta \:)}^{2} + {(1 + cot\theta \:)}^{2} = {(sec\theta \: + cosec\theta \:)}^{2}$

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

$\boxed{ \bf{ \: 1 + {tan}^{2}\theta \: = {sec}^{2} \theta \:}}$

$\boxed{ \bf{ \: 1 + {cot}^{2}\theta \: = cosec^{2}\theta \:}}$

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

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

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

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

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

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

Consider,

$\rm :\longmapsto\:{(1 + tan\theta \:)}^{2} + {(1 + cot\theta \:)}^{2}$

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

$\rm \: = \: (1 + {tan}^{2}\theta) \: + (1 + {cot}^{2}\theta) \: + 2(cot\theta \: + tan\theta \: )$

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

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

$\rm \: = \: {sec}^{2}\theta \: + {cosec}^{2}\theta \: +2\bigg(\dfrac{1}{sin\theta \:cos\theta \:} \bigg)$

$\rm \: = \: {sec}^{2}\theta \: + {cosec}^{2}\theta \: +2 \: sec\theta \:cosec\theta \:$

$\rm \: = \: {(sec\theta \: + cosec\theta \:)}^{2}$

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

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