Answers: 2
![\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1 \\\\ 1+cot^2(\theta)=csc^2(\theta) \\\\ 1+tan^2(\theta)=sec^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ tan^2(x)[1+cot^2(x)]\implies tan^2(x)[csc^2(x)]\implies \cfrac{\underline{sin^2(x)}}{cos^2(x)}\cdot \cfrac{1}{\underline{sin^2(x)}} \\\\\\ \cfrac{1}{cos^2(x)}\implies sec^2(x)](/tpl/images/0036/2359/24a22.png)
![\tan^2x\cdot(1+\cot^2x)\\\\\text{use}\ \tan x=\dfrac{\sin x}{\cos x}\ \text{and}\ \cot x=\dfrac{\cos x}{\sin x}\\\\=\left(\dfrac{\sin x}{\cos x}\right)^2\cdot\left[1+\left(\dfrac{\cos x}{\sin x}\right)^2\right]\\\\\text{use}\ \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\=\dfrac{\sin^2x}{\cos^2x}\cdot\left(1+\dfrac{\cos^2x}{\sin^2x}\right)\\\\\text{use distributive property}\\\\=\dfrac{\sin^2x}{\cos^2x}\cdot1+\dfrac{\sin^2x}{\cos^2x}\cdot\dfrac{\cos^2x}{\sin^2x}](/tpl/images/0036/2359/9612e.png)
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