If cos(α+β)=45, sin(α-β)=513 and α,β lie between 0 and π4, then tan2α=?
1663
5633
2833
None of these
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+α–β
Step 2. Take "tan" on both sides, we get
tan2α=tan(α+β+α–β)
tan2α=[tan(α+β)+tan(α–β)][1–tan(α+β)tan(α–β)] …(1) ∵tan(θ+ϕ)=tanθ+tanϕ1-tanθtanϕ
Also,
tan(α+β)=sin(α+β)cos(α+β)=3/54/5=34
tan(α–β)=sin(α–β)cos(α–β)=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(48–15)/48=5633
Hence, Option ‘B’ is Correct.