Question

if cos⁴x+cos²x=1. then prouve it that tan⁴x+tan²x=1​

Answer 1

Answer:

Note:

a). (cosX)^2 + (sinX)^2 = 1

b). 1 + (tanX)^2 = (secX)^2

c). 1 + (cotX)^2 = (cosecX)^2

d). tanX = sinX/cosX

e). cosX•secX = 1

Given:

(cosX)^4 + (cosX)^2 = 1

To prove:

(tanX)^4 + (tanX)^2 = 1

Proof:

We have;

=> (cosX)^4 + (cosX)^2 = 1

=> (cosX)^4 = 1 - (cosX)^2

=> (cosX)^4 = (sinX)^2

{using formula-(a)}

=> (cosX)^2•(cosX)^2 = (sinX)^2

=> (cosX)^2 = (sinX)^2/(cosX)^2

=> (cosX)^2 = (tanX)^2 -------(1)

Now, we have;

LHS = (tanX)^4 + (tanX)^2

= (tanX)^2{ (tanX)^2 + 1 }

= (tanX)^2(secX)^2

{using formula-(b)}

= (cosX)^2(secX)^2

= 1. {using eq-(1)}

= RHS

Hence proved.