Derive the expression 1 + tan^2x

Let us derive the equation

We know the identity sin2(x)+cos2(x)=1 ——-(i)

Dividing throughout the equation by cos2(x) We get

sin2(x)/cos2(x) + cos2(x)/cos2(x) = 1/cos2(x)

We know that

sin2(x)/cos2(x)= tan2(x), and

cos2(x)/cos2(x) = 1

So the equation (i) after substituting becomes

tan2(x) +1= 1/cos2(x) ——–(ii)

Now we know that 1/cos2(x)= sec2(x)

So on substitution equation (ii) becomes

tan2(x) +1= sec2(x)

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