Question-176211

Question Number 176211 by adhigenz last updated on 15/Sep/22

Answered by mahdipoor last updated on 15/Sep/22

2021 \frac{c o s (a + b)}{c o s a} s i n b = \frac{s i n a}{c o s a}

2021 \frac{c o s a . c o s b - s i n a . s i n b}{c o s a} s i n b = t a n a

2021 c o s b . s i n b - 2021 t a n a . {s i n}^{2} b = t a n a

\Rightarrow t a n a = \frac{2021 c o s b . s i n b}{1 + 2021 {s i n}^{2} b} = f (b)

m a x f = m a x t a n a

{\begin{cases} c o s b . s i n b = \frac{s i n 2 b}{2} \\ {s i n}^{2} b = \frac{1 - c o s 2 b}{2} \end{cases} \Rightarrow

f (b) = \frac{s i n (2 b)}{\frac{2023}{2021} - c o s (2 b)} (g e t \frac{2023}{2021} = c)

\frac{d f}{d b} = 0 \Rightarrow c o s (2 b) = \frac{1}{c} \Rightarrow s i n (2 b) = \frac{\sqrt{c^{2} - 1}}{c}

m a x f = \frac{\sqrt{c^{2} - 1} / c}{c - 1 / c} = \frac{\sqrt{c^{2} - 1}}{c^{2} - 1} = \frac{2021 \sqrt{2022}}{4044}