Question

cosec A(1+ cos A)( cosec A- cot A)=1

Answer 1

Question :

Prove that,

$\sf{cosecA (1+cosA)(cosecA-cotA)=1}$

Solution :

Taking L.H.S,

$\sf{cosecA (1+cosA)(cosecA-cotA)}$

$\implies\sf{\dfrac{1}{sinA}(1+cosA)(\dfrac{1}{sinA}-\dfrac{cosA}{sinA})}$

$\implies\sf{\dfrac{1}{sinA}(1+cosA)(\dfrac{1-cosA}{sinA})}$

$\implies\sf{\dfrac{(1+cosA)(1-cosA)}{sin^2A}}$

Use the identity : (a+b)(a-b)= a² - b²

$\implies\sf{\dfrac{1-cos^2A}{sin^2A}}$

Use the identity : sin²A = 1 - cos²A

$\implies\sf{\dfrac{sin^2A}{sin^2A}}$

$\implies\sf{1}$

L.H.S = R.H.S [ Proved]

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More identities :-

• sin²A + cos²A = 1

• 1 + tan²A = sec²A

• 1 + cot²A = cosec²A

• cos²A - sin²A = cos2A

• sinA = $\sf{\sqrt{1-cos^2A}}$

• sin2A = 2 sinA cosA

• cos2A = 2cos²A - 1

• 1 - cos2A = 2 sin²A

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Answer 2

$\huge \mathfrak{Prove \: That:-}$

$\implies$ cosec A(1+ cos A)( cosec A- cot A)=1

$\large\underline\mathrm{Solution}$

$\large\underline\mathrm{LHS}$

$\implies$ cosec A(1+ cos A)( cosec A- cot A)

$\implies$ 1/sinA (1 + cosA)(1 - cosA / sinA)

$\implies$ (1 + cosA)(1 + cosA)/sin²A. [(a + b)(a - b) = a² - b²

$\implies$ 1 + cos²A/sin²A. [(a + b)(a - b) = a² - b²

$\implies$ sin²A/sin²A. [sin²A = 1 + cos²A]

$\large\underline\mathrm{LHS \: = \: RHS. \: Proved.}$

$\huge \mathfrak{Basic \: Formula:-}$

$\implies$ sin²A + cos²A = 1

$\implies$ 1 + tan²A = sec²A

$\implies$ cos²A - sin²A = cos2A

$\implies$ sinA = √1 - cos²A

$\implies$ 1 - cos2A = 2sin²A

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