tan² A - tan² B = \(\frac { { \sin }^{ 2 }A }{ { \cos }^{ 2 }A } -\frac { { \sin }^{ 2 }B }{ { \cos }^{ 2 }B } \)
\(=\frac { { \sin }^{ 2 }A{ \cos }^{ 2 }B-{ \sin }^{ 2 }B{ \cos }^{ 2 }A }{ { \cos }^{ 2 }A{ \cos }^{ 2 }B } \)
\(=\frac{\sin ^2 A\left(1-\sin ^2 B\right)-\sin ^2 B\left(1-\sin ^2 A\right)}{\cos ^2 A \cos ^2 B}\)
\(=\frac { { \sin }^{ 2 }A-{ \sin }^{ 2 }A{ \sin }^{ 2 }B-{ \sin }^{ 2 }B+{ \sin }^{ 2 }A{ \sin }^{ 2 }B }{ { \cos }^{ 2 }A{ \cos }^{ 2 }B } =\frac { { \sin }^{ 2 }A-{ \sin }^{ 2 }B }{ { \cos }^{ 2 }A{ \cos }^{ 2 }b } \)

(cosec θ - sin θ)(sec θ - cos θ)(tan θ + cot θ)
\(=\left( \frac { 1 }{ \sin\theta } -\sin \theta \right) \left( \frac { 1 }{ \cos\theta } -\cos\theta \right) \left( \frac { \sin\theta }{ \cos\theta } +\frac { \cos\theta }{ \sin\theta } \right) \)
\(=\frac { 1-{ \sin }^{ 2 }\theta }{ \sin\theta } \times \frac { 1-{ \cos }^{ 2 }\theta }{ \cos\theta } \times \frac { { \sin }^{ 2 }\theta +{ \cos }^{ 2 }\theta }{ \sin\theta \cos\theta } \)
\(=\frac { { \cos }^{ 2 }\theta { \sin }^{ 2 }\theta \times 1 }{ { \sin }^{ 2 }\theta { \cos }^{ 2 }\theta } =1\)

\(\frac { (1+\cot A+\tan A)(\sin A-\cos A) }{ { \sec }^{ 3 }A-{ cosec }^{ 3 }A } \)
= \(\frac { \left( 1+\frac { \cos A }{ \sin A } +\frac { \sin A }{ \cos A } \right) \left( \sin A-\cos A \right) }{ (\sec A-cosecA)({ \sec }^{ 2 }A+\sec A cosecA+{ cosec }^{ 2 }A } \)
= \(\frac { \frac { (\sin A \cos A+{ \cos }^{ 2 }A+{ \sin }^{ 2 }A)(\sin A-\cos A) }{ \sin A \cos A } }{ (\sec A- cosecA)\left( \frac { 1 }{ { \cos }^{ 2 }A } +\frac { 1 }{ \cos A \sin A } +\frac { 1 }{ { \sin }^{ 2 }A } \right) } \)
= \(\frac { (\sin A \cos A+1)\left( \frac { \sin A }{ \sin A\cos A } -\frac { \cos A }{ \sin A \cos A } \right) }{ (\sec A-cosecA)\left( \frac { { \sin }^{ 2 }A+\sin A \cos A+{ \cos }^{ 2 }A }{ { \sin }^{ 2 }A{ \cos }^{ 2 }A } \right) } \)
= \(\frac { (\sin A \cos A+1)(\sec A-cosecA) }{ (\sec A-cosecA)(1+\sin A \cos A) } \times { \sin }^{ 2 }A{ \cos }^{ 2 }A={ \sin }^{ 2 }A{ \cos }^2A\)

\(\frac { \sin A }{ \sec A+\tan A-1 } +\frac { \cos A }{ cosecA+\cot A-1 } =1\)
=  \(\frac { \sin A(cosecA+\cot A-1)+\cos A(\sec A+\tan A-1) }{ (\sec A+\tan A-1)(cosec A+\cot A-1) } \)
=  \(\frac { \sin A cosecA+\sin A \cot A-\sin A+\cos A \sec A+\cos A\tan A-\cos A }{ (\sec A+\tan A-1)(cosecA+\cot A-1) } \)
= \(\frac { 1+\cos A-\sin A+1+\sin A-\cos A }{ \left( \frac { 1 }{ \cos A } +\frac { \sin A }{ \cos A } -1 \right) \left( \frac { 1 }{ \sin A } +\frac { \cos A }{ \sin A } -1 \right) } \)
= \(\frac { 2 }{ \left( \frac { 1+sinA-cosA }{ cosA } \right) \left( \frac { 1+cosA-sinA }{ sinA } \right) } \)
=  \(\frac { 2sinAcosA }{ (1+sinA-cosA)(1+cosA-sinA) } \)
=  \(\frac { 2sinAcosA }{ [1+(sinA-cosA)][1-(sinA-cosA)] } =\frac { 2sinAcosA }{ 1-{ (sinA-cosA) }^{ 2 } } \)
=  \(\frac { 2sinAcosA }{ 1-({ sin }^{ 2 }A+{ cos }^{ 2 }A-2sinAcosA) } =\frac { 2sinAcosA }{ 1-(1-2sinAcosA) } \)
= \(\frac { 2sinAcosA }{ 1-(1+2sinAcosA) } \) = \(\frac { 2sinAcosA }{ 2sinAcosA} \) = 1

\(\operatorname{cosec} \theta+\cot \theta=P\) எனக் கொடுக்கப்பட்டுள்ளது   .....(1)
\(\operatorname{cosec}^2 \theta-\cot ^2 \theta =1\)  (முற்றொருமை)
\(\operatorname{cosec} \theta-\cot \theta =\frac{1}{\operatorname{cosec} \theta+\cot \theta} \)
\(\operatorname{cosec} \theta-\cot \theta =\frac{1}{P}\)   .....(2)
(1) மற்றும் (2) ஆகியவற்றைக் கூட்டக்கிடைப்பது, \(2 \operatorname{cosec} \theta=P+\frac{1}{P}\)
\(2 \operatorname{cosec} \theta=\frac{P^2+1}{P} \quad \ldots \text { (3) }\)
(2) -லிருந்து (1) -ஐ கழித்தால் கிடைப்பது, \(2 \cot \theta=P-\frac{1}{P}\)
\(2 \cot \theta=\frac{P^2-1}{P} \quad \ldots(4)\)
(4) -ஐ (3) -ஆல் வகுக்கக் கிடைப்பது, \(\frac{2 \cot \theta}{2 \operatorname{cosec} \theta}=\frac{P^2-1}{P} \times \frac{P}{P^2+1}\)
\(\Rightarrow \cos \theta=\frac{P^2-1}{P^2+1}\) எனக் கிடைக்கிறது.