Question

prove that sin theta + cosec theta ÷ sin theta = 2+cot square theta

Answer 1

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Answer 2

GIVEN :

The trignometric equation is $\frac{sin\theta+cosec\theta}{sin\theta}=2+cot^2\theta$

TO PROVE :

The given equation $\frac{sin\theta+cosec\theta}{sin\theta}=2+cot^2\theta$ is true

SOLUTION :

To prove that LHS=RHS

From the given equation take LHS

$\frac{sin\theta+cosec\theta}{sin\theta}$

By using the trignometric identity :

$cosecx=\frac{1}{sinx}$

$=\frac{sin\theta+\frac{1}{sin\theta}}{sin\theta}$

$=\frac{sin\theta}{sin\theta}+\frac{1}{sin\theta\times sin\theta}$

$=1+\frac{1}{sin^2\theta}$

$=\frac{sin^2\theta+1}{sin^2\theta}$

By using the trignometric identity :

$sin^2x+cos^2x=1$

$=\frac{sin^2\theta+(sin^2\theta+cos^2\theta)}{sin^2\theta}$

$=\frac{sin^2\theta+sin^2\theta+cos^2\theta}{sin^2\theta}$

$=\frac{2sin^2\theta+cos^2\theta}{sin^2\theta}$

$=\frac{2sin^2\theta}{sin^2\theta}+\frac{cos^2\theta}{sin^2\theta}$

By using the trignometric identity :

$cotx=\frac{cosx}{sinx}$

$=2+cot^2\theta$=RHS

Hence LHS = RHS

∴ The given equation $\frac{sin\theta+cosec\theta}{sin\theta}=2+cot^2\theta$ is true

Hence proved.