Prove the following identities, where the angles involved are acute angles for which the expressions are defined :

(i) \( (cosec \theta - cot \theta )^2 \ = \ \frac{1 - cos \theta}{1 + cos \theta} \)

(ii) \( \frac{cosA}{1+sinA} \ + \ \frac{1+sinA}{cosA} \ = \ 2 secA \)

(iii) \( \frac{tan \theta}{1 - cot\theta} \ + \ \frac{cot \theta}{1 - tan\theta} \ = \ 1 \ + \ sec \theta cosec \theta \)

[Hint : Write the expression in terms of sin \(\theta\) and cos \(\theta\)]

(iv) \( \frac{1 \ + \ secA}{secA} \ = \ \frac{sin^2A}{1 \ - \ cosA} \)

[Hint : Simplify LHS and RHS separately]

(v) \( \frac{cosA \ - \ sinA \ + \ 1}{ cosA \ + \ sinA \ - \ 1} \ = \ cosecA \ + \ cotA \) , using the identity \( cosec^2A \ = \ 1 \ + \ cot^2A \)

(vi) \( \sqrt{ \frac{1 \ + \ sinA}{1 \ - \ sinA}} \ = \ secA \ + \ tanA \)

(vii) \( \frac{sin\theta - 2sin^3 \theta}{2cos^3 \theta - cos\theta} \ = \ tan \theta \)

(viii) \( (sinA \ + \ cosecA)^2 \ + \ (cosA \ + \ secA)^2 \ = \ 7 \ + \ tan^2A \ + \ cot^2A \)

(ix) \( (cosecA \ - \ sinA)(secA \ - \ cosA) \ = \ \frac{1}{tanA \ + \ cotA} \)

[Hint : Simplify LHS and RHS separately]

(x) \( ( \frac{1 \ + \ tan^2A}{1 \ + \ cot^2A} ) \ = \ ( \frac{1 \ - \ tanA}{1 \ - \ cotA})^2 \ = \ tan^2A \)

(i) L.H.S. = \( (cosec \theta - cot \theta )^2 \ \)
\( = \ ( \frac{1}{sin\theta} \ - \ \frac{cos \theta}{ sin \theta} )^2 \)
\( = \ ( \frac{1 \ - \ cos \theta }{ sin \theta } )^2 \)

\( = \ \frac{ ( 1 \ - \ cos \theta )^2 }{ sin^2 \theta} \ \)
\( = \ \frac{(1 \ - \ cos \theta)^2 }{ 1 \ - \ cos^2 \theta} \)
[ \(\because sin^2 \theta \ = \ 1 \ - \ cos^2 \theta \) ]

\( = \ \frac{(1 \ - \ cos\theta )^2}{(1 \ - \ cos\theta)(1 \ + \ cos\theta)} \ \)
\( = \ \frac{1 \ - \ cos\theta}{1 \ + \ cos\theta} \) = R.H.S.
[ \(\because A^2 \ - \ B^2 \ = \ (A+B)(A-B) \) ]


(ii) L.H.S. = \( \frac{cosA}{1+sinA} \ + \ \frac{1+sinA}{cosA} \)
\( = \ \frac{cos^2A \ + \ (1 \ + \ sinA)^2}{cosA(1 \ + \ sinA)} \)

\( = \ \frac{cos^2A \ + \ 1 \ + \ 2sinA \ + \ sin^2A}{cosA(1 \ + \ sinA)} \)
\( = \ \frac{(cos^2A \ + \ sin^2A) \ + \ 1 \ + \ 2sinA}{cosA(1 \ + \ sinA)} \)

\( = \ \frac{1 \ + \ 1 \ + \ 2sinA}{cosA(1 \ + \ sinA)} \)
[ \(\because sin^2A \ + \ cos^2A \ = \ 1 \) ]

\( = \ \frac{2 \ + \ 2sinA}{cosA(1 \ + \ sinA)} \ \)
\( = \ \frac{2(1 \ + \ sinA)}{cosA(1 \ + \ sinA)} \)

\( = \ \frac{2}{cosA} \ = \ 2secA \ = \) R.H.S.


(iii) L.H.S. = \( \frac{tan \theta}{1 - cot\theta} \ + \ \frac{cot \theta}{1 - tan\theta} \)
\( = \frac{ \frac{sin \theta}{cos\theta}}{ 1 \ - \ \frac{cos\theta}{sin\theta}} \ + \ \frac{ \frac{cos\theta}{sin\theta}}{1 \ - \ \frac{sin\theta}{cos\theta}} \)

\( \frac{sin^2 \theta}{cos\theta(sin\theta \ - \ cos\theta)} \ + \ \frac{cos^2 \theta}{sin\theta(cos\theta \ - \ sin\theta)} \)
\( = \ ( \frac{1}{sin\theta \ - \ cos\theta}) [ \frac{sin^2 \theta}{cos\theta} \ - \ \frac{cos^2 \theta}{sin\theta} ] \)

\( = \ \frac{sin^3 \theta \ - \ cos^3 \theta}{ cos\theta sin\theta( sin\theta \ - \ cos\theta)} \)
\( = \ \frac{(sin\theta \ - \ cos\theta)( sin^2 \theta \ + \ sin\theta cos\theta \ + \ cos^2 \theta)}{( cos\theta sin\theta( sin\theta \ - \ cos\theta)} \) [ \( \because a^3 \ - \ b^3 \ = \ (a-b)(a^2 \ + \ ab \ + \ b^2) \) ]

\( \frac{1 \ + \ sin\theta cos\theta}{sin\theta cos\theta} \)
[ \( \because sin^2 \theta \ + \ cos^2 \theta \ = \ 1 \) ]

\( = \ 1 \ + \ sec \theta cosec \theta \) = R.H.S.


(iv) L.H.S. \( = \ \frac{1 \ + \ secA}{secA} \ \)
\( = \ 1 \ + \ \frac{1}{secA} \ = \ 1 \ + \ cosA \)
\( = \ \frac{(1 \ - \ cosA )(1 \ + \ cosA)}{1 \ - \ cosA} \)

\( = \ \frac{1 \ - \ cos^2A}{1 \ - \ cosA} \)
[ \( \because A^2 \ - \ B^2 \ = \ (A+B)(A-B) \) ]

\( \frac{sin^2A}{1 \ - \ cosA} \)
[ \( \because sin^2A \ = \ 1 \ - \ cos^2A \) ]

= R.H.S.

(v) L.H.S. = \( \frac{cosA \ - \ sinA \ + \ 1}{ cosA \ + \ sinA \ - \ 1} \)
\( = \ \frac{ \frac{ cosA \ - \ sinA \ + \ 1}{sinA}}{ \frac{cosA \ + \ sinA \ - \ 1}{sinA}} \)

\( = \ \frac{cotA \ - \ 1 \ + \ cosecA}{cosA \ + \ 1 \ - \ cosecA} \)
[ \(\because 1 \ + \ cot^2A \ = \ cosec^2A \) ]

\( = \ \frac{cotA \ + \ cosecA \ - \ (cosec^2A \ - \ cot^2A)}{cotA \ - \ cosecA \ + \ 1} \)
\( = \ \frac{cotA \ + \ cosecA \ - \ (cosecA \ + \ cotA)(cosecA \ - \ cotA)}{cotA \ - \ cosecA \ + \ 1} \)
[ \(\because A^2 \ - \ B^2 \ = \ (A+B)(A-B) \) ]

Taking common(cosecA + cotA)

\( = \ \frac{(cosecA \ + \ cotA)(1 \ - \ cosecA \ + \ cot)}{(cotA \ - \ cosecA \ + \ 1 )} \)
\( = \ cosec A \ + \ cot A \) = R.H.S.


(vi) L.H.S. = \( \sqrt{ \frac{1 \ + \ sinA}{1 \ - \ sinA}} \)
\( = \ \sqrt{ \frac{1+sinA}{1-sinA} × \frac{1+sinA}{1+sinA}} \)
[Multiplying and dividing by \( \sqrt{1+sinA} \) ]

\( = \ \sqrt{ \frac{(1+sinA)^2}{1-sin^2A}} \ = \ \sqrt{ \frac{(1+sinA)^2}{cos^2A}} \)
[ \(\because sin^2A \ + \ cos^2A \ = \ 1 \) ]

\( = \ \frac{1+sinA}{cosA} \)
\( = \ \frac{1}{cosA} \ + \ \frac{sinA}{cosA} \)

\( = \ secA \ + \ tanA \)
[ \(\because tanA \ = \ \frac{sinA}{cosA} \) and \( secA \ = \ \frac{1}{cosA} \) ]

= R.H.S.


(vii) L.H.S. = \( \frac{sin\theta - 2sin^3 \theta}{2cos^3 \theta - cos\theta} \)
\( = \ \frac{ sin \theta(1-2sin^2 \theta)}{cos \theta (2cos^2 \theta -1)} \)

\( = \ tan \theta [ \frac{1-sin^2 \theta - sin^2 \theta}{ cos^2 \theta + cos^2 \theta - 1} ] \)
\( = \ tan \theta [ \frac{ cos^2 \theta - sin^2 \theta }{ cos^2 \theta - sin^2 \theta } ] \)
[ \( \because sin^2 \theta \ + \ cos^2 \theta \ = \ 1 \) ]

\( = \ tan\theta \)

= R.H.S.


(viii) L.H.S. =\( (sinA \ + \ cosecA)^2 \ + \ (cosA \ + \ secA)^2 \)
\( = \ (sin^2A \ + \ cosec^2A \ + \ 2sinAcosecA) \ + \ (cos^2A \ + \ sec^2A \ + \ 2cosAsecA) \)

\( = \ (sin^2A \ + \ cosec^2A \ + \ 2 ) \ + \ (cos^2A \ + \ sec^2A \ + \ 2 ) \)
\( = \ 1+ cosec^2A \ + \ sec^2A \ + \ 4 \) [ \( \because sin^2A \ + \ cos^2A \ = \ 1 \) ]

\( = \ 1 \ + \ cot^2A \ + \ 1 \ + \ tan^2A \ + \ 5 \) [ \( \because sec^2A \ = \ 1 \ + \ tan^2A \) and \( cosec^2A \ = \ 1 \ + \ cot^2A \) ]

\( = \ 7 \ + \ tan^2A \ + \ cot^2A \) = R.H.S.


(ix) L.H.S. = \( (cosecA \ - \ sinA)(secA \ - \ cosA) \)
\( = ( \frac{1}{sinA} \ - \ sinA)( \frac{1}{cosA} \ - \ cosA) \)

\( = \ ( \frac{1 \ - \ sin^2A}{sinA} )( \frac{1 \ - \ cos^2A}{cosA} ) \)
\( = \frac{ cos^2A}{sinA} × \frac{sin^2A}{cosA} \)

\( = \ sinA cosA \)
\( = \ \frac{sinAcosA}{sin^2A \ + \ cos^2A} \)
[ \(\because sin^2A \ + \ cos^2A \ = \ 1 \) ]

Dividing Numerator and Denominator by \( sinA cosA \), we get

\( \frac{ \frac{sinAcosA}{sinAcosA}}{ \frac{sin^2A}{sinAcosA} \ + \ \frac{cos^2A}{sinAcosA}} \)
\( = \ \frac{1}{ \frac{sinA}{cosA} \ + \ \frac{cosA}{sinA}} \)

\( = \ \frac{1}{tanA \ + \ cotA} \) = R.H.S


(x) L.H.S. = \( ( \frac{1 \ + \ tan^2A}{1 \ + \ cot^2A} ) \)
\( = \ \frac{sec^2A}{cosec^2A} \)

\( = \ \frac{1}{cos^2A} \ × \ sin^2A \ = \ tan^2A \) [ \(\because secA \ = \ \frac{1}{cosA} \) and \( cosecA \ = \ \frac{1}{sinA} \) ]

R.H.S. = \( \frac{1 \ - \ tanA}{1 \ - \ cotA})^2 \)
\( = \ ( \frac{1 \ - \ tanA}{ 1 \ - \ \frac{1}{tanA}})^2 \)

\( = \ ( \frac{1 \ - \ tanA}{ \frac{tanA \ - \ 1}{tanA}})^2 \) \( = \ tan^2A \)