Question

Prove the following trigonometrici identity (1+cot²A) sin²A=1​

Answer 1

$\bold {Identity : cosec^2A−cot^2A=1}$

$\bold{LHS =(1+cot^2A)sin^2A}$

$\bold{ =cosec^2A×sin {}^{2} A }$

$\bold{ = \frac{1}{ \sin^{2}A} \times \sin^{2}A}$

=1

$\bold { =RHS}$

Answer 2

Topic:-

Trigonometry

Question:-

$Prove \:the \:following\: trigonometrical\: identity\: (1+cot²\theta)sin²\theta=1$

Solution:-

$We \: know \: that \\ \\ {csc}^{2} \theta - {cot}^{2} \theta = 1 \\ \\ {csc }^{2} \theta = 1 + co {t}^{2} \theta \:$

$csc²\theta\:=\:1+cot²\theta\:➡1$

$So, \: now \: take \: the \: question$

$Given \: that \:LHS =(1+cot²\theta)sin²\theta$

To Find RHS

Substitute the equation➡1 in the given question

$(1+cot²\theta)sin²\theta\:=\:csc²\theta\:×sin²\theta$

We know That $sin²\theta ×csc²\theta=1$

So, RHS=1

Hence Proved\\:

More Information:-

Trigon metric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

csc²θ - cot²θ = 1

Trigometric relations

sinθ = 1/cscθ

cosθ = 1 /secθ

tanθ = 1/cotθ

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Trigonmetric ratios

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

cotθ = adj/opp

cscθ = hyp/opp

secθ = hyp/adj

Sinθ ×cscθ=1

cosθ × Secθ=1

tanθ×cotθ=1