Cot A-cosec A-1 - A-cosec A-1-1-cos a - sin A

suppose the problem is to prove that (cotA+cosecA 1)/(cotA cosecA+1)=(1+cosA)/sin A Since cosec^2 A cot^2 A = 1, LHS can be rewritten as cot A+cosec A (cos ...

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Standard X

Mathematics

Trigonometric Identities

Cot A+cosec A...

Question

Prove: $\frac{cot A + c o s e c A - 1}{cot A - c o s e c A + 1} = \frac{1 + cos A}{sin A}$

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Solution

LHS: $\frac{cot A + c o s e c A - 1}{cot A - c o s e c A + 1}$
$= \frac{cot A + c o s e c A - (c o s e c^{2} A - {cot}^{2} A)}{cot A - c o s e c A + 1}$
$= \frac{cot A + c o s e c A - (c o s e c A - cot A) (c o s e c A + cot A)}{cot A - c o s e c A + 1}$
$= \frac{(cot A + c o s e c A) (1 - c o s e c A + cot A)}{cot A - c o s e c A + 1}$
$= cot A + c o s e c A$
$= \frac{cos A}{sin A} + \frac{1}{sin A}$
$= \frac{1 + cos A}{sin A}$
$=$ RHS

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Prove:cotA+cosecA−1cotA−cosecA+1=1+cosAsinA

LHS: cotA+cosecA−1cotA−cosecA+1=cotA+cosecA−(cosec2A−cot2A)cotA−cosecA+1=cotA+cosecA−(cosecA−cotA)(cosecA+cotA)cotA−cosecA+1=(cotA+cosecA)(1−cosecA+cotA)cotA−cosecA+1=cotA+cosecA=cosAsinA+1sinA=1+cosAsinA= RHS

Prove: $\frac{cot A + c o s e c A - 1}{cot A - c o s e c A + 1} = \frac{1 + cos A}{sin A}$