Question

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$\sf {\dfrac{\sin \theta}{1-\cos\theta}=\cosec \theta+\cot \theta}$​

Answer 1

How to proceed

Here the concept of trignometry ratios, identities and rationalization of denominator is used. First we will rationalize the denominator by multiplying the L.H.S with (1+cos∅) then cancel out the sin∅ from the numerator and denominator. Apply the trigonometric ratio and solve it further to get the desired result.

Solution

L.H.S

$\sf {\dfrac{\sin \theta}{1-\cos\theta}}$

First, Rationalize the denominator by multiplying the L.H.S by (1+cos∅)

$\sf { \implies\dfrac{sin \theta}{1-\cos\theta} \times \green{ \dfrac{1 + cos \theta}{1 + cos \theta} }} \\ \\ \sf{ \implies\dfrac{sin \theta \times(1-\cos\theta )}{1 - {cos}^{2} \theta}}$

»We know that cos²∅+sin²∅ = 1

$\sf{ \implies\dfrac{sin \theta \times(1-\cos\theta )}{ \red{{sin}^{2} \theta}}}$

» Cancel sin∅ from numerator and denominator

$\sf{ \implies\dfrac{(1-\cos\theta )}{ \red{sin \theta}}} \\ \\ \sf{ \implies \frac{1}{sin \theta} - \frac{cos \theta}{sin \theta}} \\ \\ \sf{ \implies \purple{cosec \theta + cot \theta }= {R.H.S}}$

➢ Hence proved

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More to know !!

$\sf{⇝sec²∅-tan²∅ = 1}$

$\sf{⇝ cosec² ∅ = 1+ cot²∅}$

$\sf{⇝ sec∅ = \frac{1}{cos∅}}$

$\sf{⇝ tan∅ = \frac{1}{cot∅}}$

$\sf{⇝ tan∅ = \frac{sin∅}{cos∅}}$

$\sf{⇝ cosec∅ = \frac{1}{sin∅}}$

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Answer 2

Answer~

$\bigstar\sf {\dfrac{\sin \theta}{1-\cos\theta}=\cosec \theta+\cot \theta}$

➬ To prove LHS = RHS. For that lets take LHS part to be equal.

$\longrightarrow { \frac{ (\sin \theta) (1 + \cos \theta)}{1 + \cos^2 \theta }}$

$\longrightarrow{ \frac{ (\sin \theta ) (1 + \cos \theta) }{ \sin^2 \theta} }$

$\longrightarrow \frac{ \sin \theta + \sin \theta \times \cos \theta }{ \sin ^2 \theta}$

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

$\longrightarrow{ \frac{1}{ \sin \theta} + \frac{ \cos \theta }{ \sin \theta } }$

$\longrightarrow \red{ \csc \theta + \cot \theta}$

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Step 1 - Taking cos theta as lcm and multiply to numerator.

Step 2- Taking value of 1+cos sq. theta as sin sq. theta.

Step 3 - Multiply value of brackets.

Step 4 - Same as step 3 just seperate two equations.

Step 5 - Cut of the same value in division we get cosec theta+ cot theta.

Hence LHS= RHS.