Question

(cosec0 - sin0) (sec0 - cos0) (tan0- cot0 ) = 1

Answer 1

Appropriate Question :-

$\rm :\longmapsto\:(cosec\theta - sin\theta )(sec\theta - cos\theta )(tan\theta + cot\theta ) = 1$

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

Consider LHS

$\rm :\longmapsto\:(cosec\theta - sin\theta )(sec\theta - cos\theta )(tan\theta + cot\theta )$

We know,

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

and

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

On substituting these Identities, we get

$\rm \:  =  \: \bigg[\dfrac{1}{sin\theta } - sin\theta \bigg]\bigg[\dfrac{1}{cos\theta } - cos\theta \bigg]\bigg[\dfrac{sin\theta }{cos\theta } + \dfrac{cos\theta }{sin\theta } \bigg]$

$\rm \:  =  \: \bigg[\dfrac{1 - {sin}^{2} \theta }{sin\theta } \bigg]\bigg[\dfrac{1 - {cos}^{2}\theta }{cos\theta } \bigg]\bigg[\dfrac{ {sin}^{2}\theta + {cos}^{2}\theta}{sin\theta \: cos\theta } \bigg]$

We know,

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

Using this, we get

$\rm \:  =  \: \bigg[\dfrac{{cos}^{2} \theta }{sin\theta } \bigg]\bigg[\dfrac{{sin}^{2}\theta }{cos\theta } \bigg]\bigg[\dfrac{ 1}{sin\theta \: cos\theta } \bigg]$

$\rm \:  =  \: 1$

Hence,

$\red{ \boxed{ \sf{ \:(cosec\theta - sin\theta )(sec\theta - cos\theta )(tan\theta + cot\theta ) = 1}}}$

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