Prove that: (sin2A +sinA)/(1+cosA+cos2A) = tanA | Trigonometric Identities | SciPiPupi

Prove that: $\dfrac{sin2A+sinA}{1+cosA+cos2A}=tanA$

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

= $\dfrac{sin2A+sinA}{1+cosA+cos2A}$

= $\dfrac{2sinAcosA+sinA}{1+cosA+2cos²A-1}$

= $\dfrac{2sinAcosA +sinA}{2cos²A +cosA}$

= $\dfrac{sinA(2cosA +1)}{cosA(2cosA +1)}$

= $\dfrac{sinA}{cosA}$

= $tanA$

= RHS

Explanation to the above answer.

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

Step 2: We further expand the multiple angles of sin and cos using the following identities. (sin2A = 2sinAcosA) and (cos2A = 2cos²A -1). 

Step 3: On the denominator, as we expand cos2A, we get +1 and -1, which when added results 0. Now, we write the obtained expression.

Step 4: To be able to divide the numerator and denominator in algebra, we need to find the factors. So, we take the common factor in numerator as well as denominator and re-write the expression.

Step 5: After we divide the common factors in the numerator and denominator, they result 1. When multiplying the remaining factors by 1, we get sinA/cosA.

Step 6: (sinA/cosA = tanA)

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: (sin2A +sinA)/(1+cosA+cos2A) = tanA | Trigonometric Identities | SciPiPupi

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