Prove that: $\sqrt{ \dfrac{1-cos \theta}{1+cos \theta}} = cosec \theta-cot \theta$

This is a class 09 Question From Trigonometric Identities chapter of Unit Trigonometry. All the steps for the solutions are mentioned in the description below. If that's hard for you to navigate, you can always visit the facebook link given at the end of every posts. 

Solution:

Taking LHS

= $\sqrt{ \dfrac{1-cos \theta}{1+cos \theta}}$

= $\sqrt{ \dfrac{1-cos \theta}{1+cos \theta}}$x $\sqrt{ \dfrac{1-cos \theta}{1-cos \theta}}$

= $\sqrt{ \dfrac{(1-cos \theta)^2}{1 -cos^2 \theta}}$

= $\dfrac{1-cos \theta}{\sqrt{sin^2 \theta}}$

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

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

= $cosec \theta - cot \theta$

= RHS



Explanation to the above answer.

Step 1: Copying the L.H.S. from the question.

Step 2: We need to rationalize the given expression. So, we multiply the expression by such a expression that would remove the square root from our denominator. So, we use (a-b)/(a-b) as we had (a+b) in the denominator.

Step 3: In numerator, as we get (a-b)(a-b), it becomes (a-b)² and in denominator, (a-b)(a+b) becomes a²-b². 

Step 4: The square root and the square in the numerator gets cancelled. And, in denominator we write sin²B instead because we know, 1-cos²B = sin²B

Step 5: The square root and the square in the denominator gets cancelled.

Step 6: We seperated the denominator.

Step 7: 1/sinB = cosecB and cosB/sijB = cotB.


Some extra help:


The first step to solve such questions is not to panic!
When you see the question having square root then while proving the Trigonometric identities then remember you need to rationalize the denominator! Yes, the denominator has to be rationalized in most of the cases while we might need to rationalize the numerator in some cases as well!

Next step is to open the formulas and turn them into that term which can be easily cancel the square root and can be rational number!

After that, we can easily solve the question by applying Trigonometric identities and tricks!

Here we used,

cosecB = 1/ sinB and cotB = cosB / sinB


Here is the Facebook link to the solution of this question in image. 

Related Notes:

Link: Introduction To Trigonometry
Link: Values of Trigonometric Ratios
Link: Compound Angles

Question: Prove that: 2 tan(45°-A) / 1+ tan² (45°-A) = cos2A. | Trigonometric Identities | SciPiPupil


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