1.

1). 02). -13). 14). 2

Answer»

secθ $( = \sqrt {2 + \sqrt {2 + \sqrt {2 +\ldots\ldots\ldots\ldots \INFTY } } } )$

∴ secθ = $(\sqrt {2 + {\RM{sec\theta }}} )$

sec2θ = 2 + secθ

sec2θ - secθ - 2 = 0

secθ $(= \frac{{1 \pm \sqrt {1 + 8} }}{2})$

secθ $( = \frac{{1 \pm 3}}{2})$

∴ secθ = 2, -1

[secθ = -1 not possible]

∴ θ = 60° for 0 < θ <$(\frac{{\rm{\PI }}}{2})$

cosθ (1 +2cosθ) = cos60°(1 +2×cos60°)

∴ $( = \frac{1}{2}\left\{ {1 + 2 \times \frac{1}{2}} \RIGHT\} = 1)$

Shortcut: -

Secθ = 2

Cosθ $( = \frac{1}{2})$

∴ θ = 60°

cos60° (1 + 2 cos60°)

$( = \frac{1}{2}\left\{ {1 + 2 \times \frac{1}{2}} \right\})$

= 1



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