Question

(a) If $a sec θ + b tan θ = m$ and $b sec θ + a tan θ = n$ , prove that $a^{2} + n^{2} = b^{2} + m^{2}$

(b) Use the identity: $sin^{2} A + cos^{2} A = 1$ to prove that $tan^{2} A + 1 = sec^{2} A$ . Hence, find the value of $tan A$ , when $sec A = \frac{5}{3}$ , where A is an acute angle.

Approach

Given $a sec θ + b tan θ = m$ and $b sec θ + a tan θ = n$ , prove $a^{2} + n^{2} = b^{2} + m^{2}$ . Expand and use $sec^{2} θ - tan^{2} θ = 1$ to show $m^{2} - n^{2} = a^{2} - b^{2}$ .

Step-by-step working

1
$m^{2} = a^{2} sec^{2} θ + b^{2} tan^{2} θ + 2 ab sec θ tan θ$ .
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2
$n^{2} = b^{2} sec^{2} θ + a^{2} tan^{2} θ + 2 ab sec θ tan θ$ .
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3
$m^{2} - n^{2} = a^{2} (sec^{2} θ - tan^{2} θ) + b^{2} (tan^{2} θ - sec^{2} θ) = a^{2} - b^{2}$ .
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4
Rearranging, $a^{2} + n^{2} = b^{2} + m^{2}$ .
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Concepts used

trigonometric identitiesprovetansecfind value

Approach

Step-by-step working

Concepts used

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