Prove that: $tan \left ( \frac{π}{4} + \theta \right) + tan \left ( \frac{π}{4} - \theta \right) = 2\;sec2 \theta$

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,

= $tan \left ( \frac{π}{4} + \theta \right) + tan \left ( \frac{π}{4} - \theta \right)$

= $tan (45°+ \theta) + tan (45° - \theta)$

= $\frac{1+tan \theta}{1-tan \theta} +\frac{1-tan \theta}{1+tan \theta}$

= $\frac{(1+tan \theta)^2 +(1-tan \theta)^2}{1^2 -tan^2 \theta}$

= $\frac{1 +2tan \theta +tan^2 \theta +1 -2tan\theta +tan^2 \theta}{1 -tan^2 \theta}$

= $\dfrac{2(1+tan^2 \theta)}{\frac{cos^2 \theta -sin^2 \theta}{cos^2 \theta}}$

= $\dfrac{2 sec^2 \theta}{\frac{cos 2\theta}{cos \theta}}$

= $\dfrac{2 \cdot \frac{1}{cos^2 \theta}}{\frac{cos 2\theta}{cos \theta}}$

= $2 \cdot  \dfrac{1}{cos 2\theta}$

= $2 sec 2\theta$

Explanation to this solution:

Step 1: Write the LHS same from the question.

Step 2: π = 180° and π/4 = 180°/4 = 45°

Step 3: tan(45°-A) = (1-tanA)/(1+tanA) and tan (45°+A) = (1+tanA)/(1-tanA)

Step 4: Take the LHS of the two terms in the expression. In the denominator as we had (a+b)(a-b), we directly write a²-b² instead. 

Step 5: Open the formulae in the numerator. Use the following formulae to open the factors. (a+b)² = a²+2ab+b² and (a-b)² = a²-2ab+b².

Step 6: In the denominator, we use tan A = sinA/cosA. And take the LCM within one step. Also, in the numerator, we just add and subtract the terms.

Step 7: In the numerator, (1+tan² A) = sec²A. This is an important Trigonometric identity. In the denominator, cos² A-sin² A is equal to cos2A. 

Step 8: sec A= 1/cosA. We use this identity in the numerator.

Step 9: As we had (a/b)/(c/d), we get ad/bc. Since, b and d were similar terms. We get a/c.

Step 10: 1/cos2A = sec2A.


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: tan (π/4 +θ) +tan(π/4 -θ) = 2 sec2θ | Trigonometric Identities | SciPiPupil

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