Question

tan²A+cot²A+2=sec²A cosec²A​

Answer 1

$\sf\large\underline{\underline{\red{\sf Question:}}}$

• To Prove : tan²A+cot²A+2=sec²A cosec²A

$\sf\large\underline{\underline{\red{\sf Solution:}}}$

We know that,

$\sf \red\bigstar\sin^2 \theta +\cos^2\theta =1$

$\sf\red\bigstar\tan^2\theta= \sec^2\theta-1$

$\sf\red\bigstar\cot^2\theta=\csc^2\theta - 1$

Using this formulas,

L.H.S

= tan²A+cot²A+2

= (sec²A - 1) + (cosec²A - 1) + 2

= sec²A -1 + cosec²A -1 + 2

= sec²A  + cosec²A  -2 + 2

= sec²A  + cosec²A  

$=\sf \dfrac{1}{cos^2A}+\dfrac{1}{\sin^2A}$

$=\sf\dfrac{\sin^2A+\cos^2A}{\cos^2A\sin^2A}$

$=\sf\dfrac{\green{1}}{\cos^2A\sin^2A}$

= sec²A cosec²A

= R.H.S

ㅤㅤㅤㅤ$\large\green{\underbrace{\sf Hence\;proved...{\displaystyle !\,}}}$

Points to know:

• sin x = 1/ cosec x
• cos x = 1 / sec x
• tan x = sin x / cos x
• cot x = cos x/ sin x
• tan x = 1/ cot x