Prove that cos A-sin A-1 -cos A-sin A-1-cosec A- A

Cos A Sin A + 1/ Cos A + Sin A 1 dividing in numerator amp; denominator with Sin A Cot A 1 + Cosec A / Cot A Cosec A + 1 now putting 1= Cosec^2 Cot ...

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Standard X

Mathematics

Solving a Quadratic Equation by Factorization Method

Prove that co...

Question

Prove that $\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = c o s e c A + c o t A$

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Solution

$\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1}$
dividing in numerator & denominator with $s i n A$
$\frac{c o t A - 1 + c o s e c A}{c o t A - c o s e c A + 1}$
now putting $1 = c o s e c^{2} - c o t^{2}$

$= \frac{(c o t A + c o s e c A) - (c o s e c^{2} A - c o t^{2} A)}{(c o t A - c o s e c A + 1)}$
$= \frac{(c o t A + c o s e c A) - (c o s e c A + c o t A) (c o s e c A - c o t A)}{c o t A - c o s e c A - 1}$
$= \frac{(c o t A + c o s e c A) [1 - c o s e c A + c o t A)]}{(c o t A - c o s e c A + 1)}$
$= (c o t A + c o s e c A)$
RHS Proved

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\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = c o s e c A + c o t A

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\frac{c o t A + c o s e c A - 1}{c o t A - c o s e c A + 1} = \frac{1 + c o s A}{s i n A} = c o s e c θ + c o t θ = \frac{s i n A}{1 - c o s A}

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Prove thatcosA−sinA+1cosA+sinA−1=cosecA+cotA

Prove that $\frac{c o s A - s i n A + 1}{c o s A + s i n A - 1} = c o s e c A + c o t A$