Question

If sec 0 + tan0 = x, show that sin0 = (x^2-1) / (x^2 + 1)

Answer 1

Appropriate Question

$\rm :\longmapsto\:If \: sec\theta + tan\theta = x, \: show \: that \: \dfrac{ {x}^{2} - 1}{ {x}^{2} + 1} = sin\theta$

Identities Used :-

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

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

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

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

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

Given that,

$\rm :\longmapsto\:x = sec\theta + tan\theta$

Now, Consider,

$\rm :\longmapsto\:\dfrac{ {x}^{2} - 1}{ {x}^{2} + 1}$

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

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

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

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

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

$\rm \:  =  \:  \:\dfrac{tan\theta}{sec\theta}$

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

$\rm \:  =  \:  \:sin\theta$

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