Prove that: cosθ / (1-sinθ) - cosθ / (1+sinθ) = 2 tanθ | Trigonometric Identities | Sci-Pi

 

Prove that: $\dfrac{cos\theta}{1- sin\theta} - \dfrac{cos\theta}{1+sin\theta} = 2 tan\theta$

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,

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

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

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

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

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

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

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

= $\dfrac{2sin\theta cos\theta}{cos\theta \cdot cos\theta}$

= $\dfrac{2sin\theta}{cos\theta}$

= $2 tan\theta$

= RHS

Explanation to the above answer.

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

Step 2: To do further subtraction, we need like terms. But, the terms are unlike. So, we take the LCM of these two terms and make them like terms.

Step 3: We write the numerator in the factor form. While, we get the denominators in the form of (a+b)(a-b). So, we write them as a²-b².

Step 4: (sin $\theta$)² means sin²$\theta$.

Step 5: We get (1 -sin²$\theta$) on the denominator. We have an identity cos²$\theta$ =  1 -sin²$\theta$. So, we write this.

Step 6: Now, we perform the subtraction between these two terms and write a single denominator.

Step 7: We had factors in the numerator. Now, we multiply the factors and re-write the expression.

Step 8: The like terms of opposite sign get added and result zero. Remaining like terms with same signs also get added.

Step 9: cos $\theta$ in the numerator and denominator get divided and result 1. Then the remaining expression is multiplied by 1, and we write the obtained expression.

Step 10: sin$\theta$ / cos$\theta$ = tan$\theta$

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: cosθ / (1-sinθ) - cosθ / (1+sinθ) = 2 tanθ  | Trigonometric Identities | Sci-Pi

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