## Trigonometry (11th Edition) Clone

$$\frac{\csc\theta\sec\theta}{\cot\theta}=\sec^2\theta$$
$$A=\frac{\csc\theta\sec\theta}{\cot\theta}$$ $$A=(\csc\theta\sec\theta)\times(\frac{1}{\cot\theta})$$ - Reciprocal Identities: $$\csc\theta=\frac{1}{\sin\theta}\hspace{1cm}\sec\theta=\frac{1}{\cos\theta}$$ So, $$\csc\theta\sec\theta=\frac{1}{\sin\theta}\frac{1}{\cos\theta}=\frac{1}{\sin\theta\cos\theta}\hspace{1cm}(1)$$ - Another reciprocal identity: $$\cot\theta=\frac{1}{\tan\theta}$$ which means $$\tan\theta=\frac{1}{\cot\theta}$$ Yet, according to a Quotient Identity: $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ Therefore, $$\frac{\sin\theta}{\cos\theta}=\frac{1}{\cot\theta}\hspace{1cm}(2)$$ Combine $(1)$ and $(2)$ back into $A$, we have $$A=\frac{1}{\sin\theta\cos\theta}\times\frac{\sin\theta}{\cos\theta}$$ $$A=\frac{1}{\cos^2\theta}$$ $$A=\sec^2\theta\hspace{1.5cm}\text{(Reciprocal Identity)}$$