Question

Use basic identities to simplify the expression.

(1/cot^2 θ) + sec θ cos θ


csc2θ

1

tan2θ

sec2θ

Answer 1

What is the reciprocal of cot?

Answer 2

\[\cot(x)=\frac{ 1 }{ \tan(x) } \implies \frac{ 1 }{ \cot(x) }=\tan(x)\]This means that we can replace 1/cot^2(theta) with just tan^2(theta). So now the expression looks something like this:\[\tan^2(\theta) + \sec(\theta)\cos(\theta)\] We can simplify this further from our knowledge of sec(x). \[\sec(x) =\frac{ 1 }{ \cos(x) }\] We can now replace the sec(x) with 1/cos(x) in the expression and we notice that the cos(x) cancels out.\[\tan^2(\theta)+\sec(\theta)\cos(\theta)=\tan^2(\theta)+\frac{ 1 }{ \cancel{\cos(\theta)} }\cancel{\cos(\theta)}=\tan^2(\theta)+1\]Now from our knowledge for tangent/secant identities, we know the following:\[\sec^2(x)=\tan^2(x)+1\] From this identity, we can conclude that:\[\tan^2(\theta) + 1 = \sec^2(\theta)\] And that's the answer.

@aga3t