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If cos α cosh \( \beta =\frac{x}{2},\sin \alpha sinh \beta =\frac{y}{2}, \)/ Prove that \( sec\left ( \alpha -i\beta \right )+sec\left ( \alpha +i\beta \right )= \frac{4x}{x^2+y^2}\)/
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written 3.0 years ago by
teamques10
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We have
$sec(\alpha + i\beta) = \dfrac{1}{cos(\alpha + i\beta)}$
=$ \dfrac{1}{cos\alpha cosi\beta + sin\alpha sini\beta}$
=$ \dfrac{1}{cos\alpha h\beta + isin\alpha sinh\beta}$
= $\dfrac{1}{(\dfrac{x}{2}) + i(\dfrac{y}{2})}$………………{by given data}
= $\dfrac{2}{x+iy}$..................................(1)
Similarly
$sec(\alpha – i\beta) =\dfrac{1}{cos(\alpha – i\beta)}$
=$ \dfrac{1}{cos\alpha cosi\beta – sin\alpha sini\beta}$
=$ \dfrac{1}{ cos\alpha cosh\beta – sin\alpha sinh\beta }$ …
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