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Prove that: $\dfrac{2 tan(45°-A)}{1 +tan^2(45°-A)} = cos2A$

This is a class 10 Question From Values of Trigonometric Ratios 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,

= $\frac{2 tan(45°-A)}{1 +tan^2(45°-A)}$

= $\dfrac{\dfrac{2(1-tanA)}{1+tanA}}{1 + \left ( \dfrac{1-tanA}{1+tanA} \right )^2}$

= $\dfrac{\dfrac{2(1-tanA)}{1+tanA}}{\dfrac{(1+tanA)^2 \;+\; (1-tanA)^2}{(1+tanA)^2}}$

= $\dfrac{ 2(1-tanA) \;\cdot (1+tanA)^2}{(1+tanA) \cdot \{(1+tanA)^2+(1-tanA)^2\}}$

= $\dfrac{ 2(1-tanA) \;\cdot (1+tanA)}{(1+tanA)^2+(1-tanA)^2\}}$

= $\dfrac{ 2(1-tan^2 A)}{2(1+tan^2A+tanA-tanA)}$

= $\dfrac{1 -tan^2A}{1+tan^2 A}$

= $\dfrac{1- \frac{sin^2A}{cos^2A}}{1+\frac {sin^2A}{cos^2A}}$

= $\dfrac{ \dfrac{cos^2A-sin^2A}{cos^2A}}{\dfrac {cos^2A+sin^2A}{cos^2A}}$

= $\dfrac{cos^2A-sin^2A}{cos^2A+sin^2A}$

= $\dfrac{cos2A}{1}$

= $cos2A$

= RHS


Explanation to the above answer.

Step 1: Taking the LHS and writing it same as in the question. 

Step 2: We have the formula of tan(45°-A) which is equal to (1-tanA)/(1+tanA). We write this in place of tan(45°-A).

Step 3: Taking the LCM of the terms in the denominator.

Step 4: Since we have a ratio in the form of (a/b)/(c/d). We can write this as (a*d)/(b*c). We re-write the expression in this way. 

Step 5:  (1+tanA) in the numerator and the denominator gets cancelled as they were in division form. (1+tanA) still remains in the numerator because it was in the form of squares. 

Step 6: In the numerator, we have (a-b)(a+b) which is equal to (a²-b²). In the denominator, we have (a-b)² which is equal to (a²-2ab+b²) and (a+b)² which is equal to (a²+2ab+b²). While adding, we can write as 2(a²+b²-ab+ab).

Step 7: Perform the simple mathematics in this step.

Step 8: Write tan²A as (sin²A/cos²A).

Step 9: Take the LCM of the terms. 

Step 10: cos²A is in b and d when compared to (a/b)/(c/d). This can be written as (a*d)/(b*c). So, they get divided and result 1.

Step 11: cos²A+sin²A = 1 and cos²A-sin²A = cos 2A.

Step 12: Write cos2A as our answer because any expression divided by 1 is equal to the same expression.



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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