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Proof by mathematical induction

If 0 < β < ...

Question

If $0 < β < α / 4, c o s (α + β) = 3 / 5 a n d cos (α - β) = 4 / 5$ find $s i n β α$ .

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Solution

$\begin{matrix} cos (α + β) = \frac{3}{5} \Rightarrow sin (α + β) = \frac{4}{5} . . . . . . . (i) c o s (α - β) = \frac{4}{5} sin (α - β) = cos (α + β) . . . . . (i i i) cos (α + β) = sin (90 - (α + β)) . . . . . . (i v) s u b (i v) i n (i i i) sin (α + β) = sin (90 - (α - β)) α + β = 90 - (α - β) β α = 90 sin (β α) = sin 90 \end{matrix}$

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0 < β < α < \frac{π}{4}, cos (α + β) = \frac{3}{5}

cos (α - β) = \frac{4}{5}

sin 2 α

0 < α, β < \frac{π}{4}, cos (α + β) = \frac{4}{5}, sin (α - β) = \frac{5}{13}

tan 2 α =

cos (α + β) = 4 / 5, sin (α - β) = 5 / 13

α, β

0

\frac{π}{4}

tan 2 α

cos (α + β) = \frac{3}{5}, sin (α - β) = \frac{5}{13}

0 < α, β < \frac{π}{4},

tan (2 α)

cos (α + β) = \frac{4}{5}, sin (α - β) = \frac{5}{13}

α, β

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\frac{π}{4}

tan 2 α =

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