Prove the following identity-sin A -1-cos A -1-cos A -sin A -2 cosec A -3 MARKS-

Formula: 1 Mark Proof: 2 Marks sin A/1+cos A + 1+cos A/sin A =sin^2 A + (1+cos A)^2/(1+cos A)sin A =sin^2 A + 1+cos^2 A+2 cos A/(1+cos A)sin A =1+1+2 cos A/(1+ ...

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Standard XII

Chemistry

Introduction to Haloalkanes

Prove the fol...

Question

Prove the following identity:
$\frac{s i n A}{1 + c o s A} + \frac{1 + c o s A}{s i n A} = 2 c o s e c A$ [3 MARKS]

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Solution

Formula: 1 Mark
Proof: 2 Marks

$\frac{s i n A}{1 + c o s A} + \frac{1 + c o s A}{s i n A}$
$= \frac{s i n^{2} A + (1 + c o s A)^{2}}{(1 + c o s A) s i n A}$
$= \frac{s i n^{2} A + 1 + c o s^{2} A + 2 c o s A}{(1 + c o s A) s i n A}$
$= \frac{1 + 1 + 2 c o s A}{(1 + c o s A) s i n A} = \frac{2 (1 + c o s A)}{(1 + c o s A) s i n A}$
$= \frac{2}{s i n A} = 2 c o s e c A$ = RHS
Hence, proved.

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Q. Prove the identity:

\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A} = \frac{2}{1 - 2 c o s^{2} A}

Q. Prove the following identity.

\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A} = \frac{2}{2 s i n^{2} A - 1}

Q. Prove that:

\frac{sin A + cos A}{sin A - cos A} + \frac{sin A - cos A}{sin A + cos A} = \frac{2}{2 {sin}^{2} A - 1}

Prove the following trigonometric identities:

(i) $\sqrt{\frac{1 + s i n A}{1 - s i n A}} = s e c A + t a n A$

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Prove the identity:
$\frac{s i n A}{1 + c o s A} = c o s e c A - c o t A$ [3 MARKS]

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Prove the following identity: sin A1+cos A+1+cos Asin A=2 cosec A [3 MARKS]

Formula: 1 Mark Proof: 2 Marks sin A1+cos A+1+cos Asin A =sin2A+(1+cos A)2(1+cos A)sin A =sin2A+1+cos2A+2 cos A(1+cos A)sin A =1+1+2 cos A(1+cos A)sin A=2(1+cos A)(1+cos A)sin A =2sin A=2 cosec A = RHS Hence, proved.

Prove the following identity:
$\frac{s i n A}{1 + c o s A} + \frac{1 + c o s A}{s i n A} = 2 c o s e c A$ [3 MARKS]