IF cos(alpha + beta) = 4/5 and sin(alpha - beta) = 5/13

FIND THE VALUE OF tan 2 alpha, where 0 is less than equal to alpha is less than equal to pie/4.

plz reply!!

Question

IF cos(alpha + beta) = 4/5 and sin(alpha - beta) = 5/13
FIND THE VALUE OF tan 2 alpha, where 0 is less than equal to
alpha is less than equal to pie/4
plz reply!!

Ankush Jain · Accepted Answer

Given, $\cos (α + β) = \frac{4}{5} a n d \sin (α - β) = \frac{5}{13} \Rightarrow \cos (α + β) = \frac{4}{5} = \frac{B}{H} a n d \sin (α - β) = \frac{5}{13} \Rightarrow \sin (α + β) = \frac{3}{5} = \frac{P}{H} [u s e p y t h a g o r a s t h e o r e m t o f i n d p e r p e n d i c u l a r] \Rightarrow (α + β) = \sin^{- 1} \frac{3}{5} (1) a n d α - β = \sin^{- 1} \frac{5}{13} (2)$ On adding (1) and (2), we get $2 α = \sin^{- 1} \frac{3}{5} + \sin^{- 1} \frac{5}{13} \Rightarrow 2 α = \sin^{- 1} {\frac{3}{5} \sqrt{1 - {(\frac{5}{13})}^{2}} + \frac{5}{13} \sqrt{1 - {(\frac{3}{5})}^{2}}} \Rightarrow 2 α = \sin^{- 1} {\frac{3}{5} \sqrt{\frac{144}{169}} + \frac{5}{13} \sqrt{\frac{16}{25}}} \Rightarrow 2 α = \sin^{- 1} {\frac{3}{5} \times \frac{12}{13} + \frac{5}{13} \times \frac{4}{5}} = \sin^{- 1} {\frac{56}{65}} \Rightarrow \sin 2 α = \frac{56}{65} = \frac{P}{H} N o w, B^{2} = H^{2} - P^{2} = 65^{2} - 56^{2} = 4225 - 3136 = 1089 \Rightarrow B = \sqrt{1089} = 33 N o w, \tan 2 α = \frac{P}{B} = \frac{56}{33}$

IF cos(alpha + beta) = 4/5 and sin(alpha - beta) = 5/13 FIND THE VALUE OF tan 2 alpha, where 0 is less than equal to alpha is less than equal to pie/4. plz reply!!

IF cos(alpha + beta) = 4/5 and sin(alpha - beta) = 5/13

FIND THE VALUE OF tan 2 alpha, where 0 is less than equal to alpha is less than equal to pie/4.

plz reply!!