Question

if cot^4 theta -cot ^2 theta =1 then proved sec ^4 theta - sec ^2 theta = 1

Solve it , please!​

Answer 1

Step-by-step explanation:

given that,

cot^4A - cot^2A = 1

to prove that,

sec^4A -sec^2A =1

so,

cot^4A - cot^2A=1

1/tan^4A -1/tan^2A =1

1 - tan^2A =tan^4A

1- (sec^2A - 1 ) = ( sec^2A - 1 )^2

1-sec^2A + 1 = sec^4A + 1 -2 sec^A

2-1=sec^4A - sec^2A

sec^4A - sec^2A = 1

hence proved

Answer 2

$cot^4(\theta )-cot^2(\theta ) = 1\\\\cot^2[cot^2(\theta) -1] =1\\\\cot^2(\theta)-1=\frac{1}{cot^2(\theta)}\\\\cot^2(\theta) = 1 + \frac{1}{cot^2(\theta )} - - - - - - - - - - - - - Equation-1\\\\\\1+tan^2(\theta) = sec^2(\theta) ------------- (Identity)\\\\But,\\\\tan^2(\theta) = \frac{1}{cot^2(\theta)} \\\\Therefore,\\\\1+cot^2(\theta) = sec^2(\theta) - - - - - - -- -----(Equation-2)$

From Equations 1 and 2:

$sec^2(\theta)=cot^2(\theta)\\\\Therefore,\\\\sec^4(\theta)-sec^2(\theta)=cot^4(\theta)-cot^2(\theta)=1$

P.S If you can link from where or which book you got this answer, it will be really helpful.