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Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

Prove that :1...

Question

Prove that : $(1 + c o t θ - c o s e c θ) (1 + t a n θ + s e c θ) = 2$

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Solution

$1 + c o t θ - c s c θ = 1 + \frac{c o s θ}{s i n θ} - \frac{1}{s i n θ}$
$= 1 + \frac{(c o s θ - 1)}{s i n θ}$
$= \frac{(s i n θ + c o s θ + 1)}{s i n θ}$
$1 + t a n θ + s e c θ = 1 + \frac{s i n θ}{c o s θ} + \frac{1}{c o s θ}$
$= \frac{(s i n θ + c o s θ + 1)}{c o s θ}$
Thus,
$(1 + c o t θ - c o s e c θ) (1 + t a n θ + s e c θ)$
$= \frac{[(s i n θ + c o s θ) - 1]}{s i n θ} * \frac{[(s i n θ + c o s θ) + 1]}{c o s θ}$
$= \frac{[(s i n θ + c o s θ)^{2} - 1]}{s i n θ c o s s i n θ c o s θ}$
$= \frac{(s i n^{2} θ + 2 s i n θ c o s θ + c o s^{2} θ - 1)}{s i n θ c o s θ}$
$= \frac{[2 s i n θ c o s θ + (s i n^{2} θ + c o s^{2} θ) - 1]}{s i n θ c o s θ}$
$= \frac{(2 s i n θ c o s θ + (1) - 1)}{s i n θ c o s θ}$
$= \frac{(2 s i n θ c o s θ)}{s i n θ c o s θ}$
= 2

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Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

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