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Question

If cos(α+β)=45, sin(α-β)=513 and α,β lie between 0 and π4, then tan2α=?


A

1663

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B

5633

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C

2833

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D

None of these

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Solution

The correct option is B

5633


Explanation for the correct option:

Step 1. Find the value of tan2α:

Given, cos(α+β)=45

sin(α+β)=35

sin(α-β)=513

cos(α-β)=1213

Now, we can write

2α=α+β+αβ

Step 2. Take "tan" on both sides, we get

tan2α=tan(α+β+αβ)

tan2α=[tan(α+β)+tan(αβ)][1tan(α+β)tan(αβ)] …(1) tan(θ+ϕ)=tanθ+tanϕ1-tanθtanϕ

Also,

tan(α+β)=sin(α+β)cos(α+β)=3/54/5=34

tan(αβ)=sin(αβ)cos(αβ)=5/1312/13=512

Step 3. Put these values in equation (1), we get

tan2α=(3/4)+(5/12)1(3/4)(5/12)=(9+5)/12(4815)/48=5633

Hence, Option ‘B’ is Correct.


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