Question

solve fast fast please
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Answer 1

Answer:

Recall 1 + tan²θ = sec²θ.

So 1 = sec²θ - tan²θ.

Therefore:

 1 + 4k² = 4k sec θ

=> sec²θ - tan²θ + 4k² = 4k sec θ

=> sec²θ - 4k sec θ + 4k² = tan²θ

=> ( sec θ - 2k )² = tan²θ

=> sec θ - 2k = ±tan θ

=> sec θ ± tan θ = 2k

If sec θ + tan θ = 2k, then we have the required result.

If sec θ - tan θ = 2k, then

sec θ + tan θ = ( sec²θ - tan²θ ) / ( sec θ - tan θ ) = 1 / 2k, and again we have the required result.

Answer 2

$\sec( \theta) = \frac{1 + 4k^{2}}{4k}$

by pythagoras theorem