If cos(α+β)=(4/5), sin (α-β)=(5/13) and α, β lie between 0 and π/4 find tan 2α

Question

If cos(α+β)=(4/5), sin (α-β)=(5/13) and α, β lie between 0 and π/4 find tan 2α (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Since, cos(α+β)=(4/5) and sin(α+β)=(5/13) ∴ tan(α+β)=(3/4)and tan (α-β)=(5/12) Now, tan 2 α = tan [(α + β) + (α - β)] =(tan (α + β)+ tan (α -β )/1 - tan ( α + β) tan ( α - β ))=((3/4)+(5/12)/1-(3)4.(5)12=(56/33)

Q. If cos(α+β)=54​,sin(α−β)=135​andα,βliebetween0andπ/4findtan2α

Since, cos(α+β)=54​ andsin(α+β)=135​ ∴tan(α+β)=43​andtan(α−β)=125​ Now, tan 2 \alpha = tan [(\alpha + \beta) + (\alpha - \beta)] =1−tan(α+β)tan(α−β)tan(α+β)+tan(α−β)​=1−43​.125​43​+125​​=3356​

Q. If $cos (α + β) = \frac{4}{5}, s in (α - β) = \frac{5}{13} an d α, βl i e b e tw ee n 0 an d π /4 f in d t an 2 α$