Prove that: (cotA -tanA) / (cotA +tanA) = cos 2A. | Trigonometric Identities | SciPiPupil

Prove that: $\dfrac{cotA-tanA}{cotA+tanA}\;=\;cos2A$

This is a class 10 Question From Multiple Angles 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 L.H.S.

= $\dfrac{cotA-tanA}{cotA+tanA}$

= $\dfrac{\;\dfrac{cosA}{sinA}\;-\;\dfrac{sinA}{cosA}\;}{\dfrac{\;cosA}{sinA}\;+\;\dfrac{sinA}{sinA}\;}$

= $\dfrac{\;\dfrac{cos²A-sin²A}{cosAsinA}\;}{\dfrac{\;cos²A+sin²A}{cosAsinA}\;}$

= $\dfrac{cos²A-sin²A}{cos²A+sin²A}$

= $\dfrac{cos²A-sin²A}{1}$

= $cos²A\;-\;sin²A$

= $cos2A$

= R.H.S.

Explanation to the above answer.

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

Step 2: Writing cosA and tanA in terms of sinA and cosA so that we can carry on further addition or subtraction. (cotA = cosA/sinA) and (tanA = sinA/cosA)

Step 3: Taking LCM of the terms each in the numerator and the denominator and performing addition and subtraction of terms, according to the assigned operators.

Step 4: In ratio a/b ÷ c/d, we get, (a*d)/(b*c). In our expression, b = d, Therefore, we get the ratio a/c.

Step 5: We use the Trigonometric identity: sin²A +cos²A = 1.

Step 6: Any term divided by 1 results the same term. So, we write (cos²A -sin²A).

Step 7: Finally, we replace the value 'cos²A -sin²A' with cos2A (Multiple Angle Formula).

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: (cotA -tanA) / (cotA +tanA) = cos 2A. | Trigonometric Identities | SciPiPupil

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