Question

If tan0 = (1 - e), show that sec 0 + tano.cosec 0 = (2-e?)32
III.​

Answer 1

$\begin{gathered}\begin{gathered}\bf\: Given-\begin{cases} &\sf{ {tan}^{2}\theta = 1 - {e}^{2} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\:show - \begin{cases} &\sf{sec\theta + {tan}^{3}\theta \: cosec\theta={\bigg( 2 - {e}^{2} \bigg) }^{\dfrac{3}{2} }}\end{cases}\end{gathered}\end{gathered}$

Identities Used :-

$\boxed{\red{\sf\: {sin}^{2}x + {cos}^{2}x = 1}}$

$\boxed{\red{\sf\:secx = \dfrac{1}{cosx}}}$

$\boxed{\red{\sf\:cosecx = \dfrac{1}{sinx}}}$

$\boxed{\red{\sf\:cotx = \dfrac{cosx}{sinx}}}$

$\boxed{\red{\sf\:tanx = \dfrac{sinx}{cosx}}}$

Let's solve the problem now!!

Given that,

$\rm :\longmapsto\: {tan}^{2}\theta = 1 - {e}^{2} - - - (1)$

Now,

Consider,

$\rm :\longmapsto\:sec\theta + {tan}^{3}\theta \: cosec\theta$

$\: \sf \: \: = \: \: \dfrac{1}{cos\theta} + \dfrac{ {sin}^{3}\theta }{ {cos}^{3} \theta} \times \dfrac{1}{sin\theta}$

$\: \sf \: \: = \: \: \dfrac{1}{cos\theta} + \dfrac{ {sin}^{2}\theta }{ {cos}^{3} \theta}$

$\: \sf \: \: = \: \: \dfrac{ {cos}^{2}\theta + {sin}^{2}\theta }{ {cos}^{3} \theta}$

$\: \sf \: \: = \: \: \dfrac{ 1}{ {cos}^{3} \theta}$

$\: \sf \: \: = \: \: {sec}^{3}\theta$

$\: \sf \: \: = \: \: {\bigg( {sec}^{2}\theta \bigg) }^{\dfrac{3}{2} }$

$\: \sf \: \: = \: \: {\bigg(1 + {tan}^{2}\theta \bigg) }^{\dfrac{3}{2} }$

$\: \sf \: \: = \: \: {\bigg(1 +1 - {e}^{2}\bigg) }^{\dfrac{3}{2} }$

$\: \sf \: \: = \: \: {\bigg(2 - {e}^{2}\bigg) }^{\dfrac{3}{2} }$

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

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