Question

cotθ + Cosecθ -1 / cotθ - cosecθ +1 = ??
Try and answer when done

Answer 1

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

Given Trigonometric function is

$\rm :\longmapsto\:\dfrac{cot \theta + cosec \theta - 1}{cot \theta - cosec \theta + 1}$

We know,

$\boxed{ \tt{ \: {cosec}^{2}x - {cot}^{2}x = 1 \: }}$

So, replacing 1 in numerator by this identity, we get

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

We know

$\boxed{ \tt{ \: {x}^{2} - {y}^{2} = (x + y)(x - y) \: }}$

$\rm \: = \:\dfrac{(cot \theta + cosec \theta ) - (cosec \theta + cot \theta )(cosec \theta - cot \theta )}{cot \theta - cosec \theta + 1}$

$\rm \: = \:\dfrac{(cot \theta + cosec \theta )\bigg[1 - (cosec \theta - cot \theta )\bigg]}{cot \theta - cosec \theta + 1}$

$\rm \: = \:\dfrac{(cot \theta + cosec \theta )\bigg[1 - cosec \theta + cot \theta\bigg]}{cot \theta - cosec \theta + 1}$

can be re-arranged as

$\rm \: = \:\dfrac{(cot \theta + cosec \theta ) \: \: \cancel{\bigg[cot \theta - cosec \theta + 1\bigg]}}{ \cancel{cot \theta - cosec \theta + 1}}$

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

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

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

Thus,

$\red{\sf :\longmapsto\:\boxed{ \tt{ \: \dfrac{cot \theta + cosec \theta - 1}{cot \theta - cosec \theta + 1} = \frac{1 + cos \theta }{sin \theta } \: }}}$

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