Prove that: cot2A +tanA = cosec2A. | Trigonometric Identities | SciPiPupil

Prove that: $cot2A +tanA = cosec2A$

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 LHS

= $cot2A +tanA$

= $\dfrac{cos2A}{sin2A} + \dfrac{sinA}{cosA}$

= $\dfrac{cos2A \cdot cosA + sinA\cdot sin2A}{sin2A \cdot cosA}$

= $\dfrac{(cos^2A-sin^2A)cosA + sinA(2sinAcosA)}{(2sinAcosA)cosA}$

= $\dfrac{cos^3A -sin^2AcosA +2sin^2AcosA}{2sinAcos^2A}$

= $\dfrac{cos^3A+sin^2AcosA}{2sinAcos^2A}$

= $\dfrac{cosA(cos^2A+sin^2A)}{2sinAcos^2A}$

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

= $\dfrac{1}{2sinAcosA}$

= $\dfrac{1}{sin2A}$

= $cosec2A$

= RHS

Explanation to the above answer.

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

Step 2: We write cot and tan in the form of sin and cos to be able to perform further addition. (cot2A = cos2A/sin2A) and (tanA = sinA/cosA). 

Step 3: We take the LCM of the two terms and add the numerator. 

Step 4: We expand the formula of multiple angles. (cos2A = cos²A -sin²A) and (sin2A = 2sinAcosA).

Step 5: We multiply the factors and write the obtained expressions.

Step 6: We have like terms in the numerator, so we add those two terms. 

Step 7: We had cosA factor common in both two terms in the numerator. So, we re-write the expression with cosA as a common factor.

Step 8: cosA is also present in the denominator. So, we divide the cosA in the numerator by the cosA in the denominator.

Step 9: (cos²A +sin²A = 1). An important Trigonometry identity.

Step 10: (2sinAcosA = sin2A). Important formula for multiple angle of sin.

Step 11: We know, 1/sinA = cosecA. So, 1/sin2A = cosec2A.

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: cot2A +tanA = cosec2A. | Trigonometric Identities | SciPiPupil

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