Question

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Answer 1

Question :

Prove that ,

$\sf{\dfrac{1+sin\:A}{cosec\:A-cot\:A}-\dfrac{1-sin\:A}{cosec\:A+cot\:A}=2(1+cot\:A)}$

Solution :

Taking L.H.S ,

$\sf{\dfrac{1+sin\:A}{cosec\:A-cot\:A}-\dfrac{1-sin\:A}{cosec\:A+cot\:A}}$

$\to\sf{\dfrac{(1+sin\:A)(cosec\:A+cot\:A)-(1-sin\:A)(cosec\:A-cot\:A)}{(cosec\:A-cot\:A)(cosec\:A+cot\:A)}}$

$\\ \to\sf{\dfrac{(cosec\:A+cot\:A+sin\:A\:\:cosec\:A+sin\:A\:\: cot\:A)}{cosec^2A-cot^2A}} \\ \sf - \frac{-(cosec\:A-cot\:A-sin\:A\:\:cosec\:A+sin\:A\:\:cot\:A}{cosec^2A-cot^2A}$

$\\ \to\sf{\dfrac{(cosec\:A+cot\:A+sin\:A\times\dfrac{1}{sin\:A}+sin\:A\times\dfrac{cos\:A}{sin\:A})}{1}} \\ - \sf \frac{(cosec\:A-cot\:A-sin\:A\times\dfrac{1}{sin\:A}+sin\:A\times\dfrac{cos\:A}{sin\:A})}{1}$

$\\ \to\sf{(cosec\:A+cot\:A+1+cos\:A)-(cosec\:A-cot\:A-1+cos\:A)}$

$\to\sf{cosec\:A+cot\:A+1+cos\:A-cosec\:A+cot\:A+1-cos\:A}$

$\to\sf{2+2cot\:A}$

$\to\sf{2(1+cot\:A)}$

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

_________________

Some identities :-

★ sin²A + cos²A = 1

★ 1 + tan²A = sec²A

★ 1 + cot²A = cosec²A

★ cos²A - sin²A = cos2A

Answer 2

Prove that ,

\sf{\dfrac{1+sin\:A}{cosec\:A-cot\:A}-\dfrac{1-sin\:A}{cosec\:A+cot\:A}=2(1+cot\:A)}

cosecA−cotA

1+sinA

−

cosecA+cotA

1−sinA

=2(1+cotA)

Solution :

Taking L.H.S ,

\sf{\dfrac{1+sin\:A}{cosec\:A-cot\:A}-\dfrac{1-sin\:A}{cosec\:A+cot\:A}}

cosecA−cotA

1+sinA

−

cosecA+cotA

1−sinA

\to\sf{\dfrac{(1+sin\:A)(cosec\:A+cot\:A)-(1-sin\:A)(cosec\:A-cot\:A)}{(cosec\:A-cot\:A)(cosec\:A+cot\:A)}}→

(cosecA−cotA)(cosecA+cotA)

(1+sinA)(cosecA+cotA)−(1−sinA)(cosecA−cotA)

\begin{gathered}\\ \to\sf{\dfrac{(cosec\:A+cot\:A+sin\:A\:\:cosec\:A+sin\:A\:\: cot\:A)}{cosec^2A-cot^2A}} \\ \sf - \frac{-(cosec\:A-cot\:A-sin\:A\:\:cosec\:A+sin\:A\:\:cot\:A}{cosec^2A-cot^2A}\end{gathered}

→

cosec

2

A−cot

2

A

(cosecA+cotA+sinAcosecA+sinAcotA)

−

cosec

2

A−cot

2

A

−(cosecA−cotA−sinAcosecA+sinAcotA

\begin{gathered}\\ \to\sf{\dfrac{(cosec\:A+cot\:A+sin\:A\times\dfrac{1}{sin\:A}+sin\:A\times\dfrac{cos\:A}{sin\:A})}{1}} \\ - \sf \frac{(cosec\:A-cot\:A-sin\:A\times\dfrac{1}{sin\:A}+sin\:A\times\dfrac{cos\:A}{sin\:A})}{1}\end{gathered}

→

1

(cosecA+cotA+sinA×

sinA

1

+sinA×

sinA

cosA

)

−

1

(cosecA−cotA−sinA×

sinA

1

+sinA×

sinA

cosA

)

\begin{gathered}\\ \to\sf{(cosec\:A+cot\:A+1+cos\:A)-(cosec\:A-cot\:A-1+cos\:A)}\end{gathered}

→(cosecA+cotA+1+cosA)−(cosecA−cotA−1+cosA)

\to\sf{cosec\:A+cot\:A+1+cos\:A-cosec\:A+cot\:A+1-cos\:A}→cosecA+cotA+1+cosA−cosecA+cotA+1−cosA

\to\sf{2+2cot\:A}→2+2cotA

\to\sf{2(1+cot\:A)}→2(1+cotA)

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

_________________

Some identities :-

★ sin²A + cos²A = 1

★ 1 + tan²A = sec²A

★ 1 + cot²A = cosec²A

★ cos²A - sin²A = cos2a

hope

it

help

.