- Demonstrate an understanding of the fundamental identities
- Demonstrate the ability to verify a trigonometric identity
#1:
Instructions: verify each identity.
$$a)\hspace{.1em}\text{cot}^2 θ + 1=\text{sec}^2 θ \hspace{.1em}\text{cot}^2 θ$$
$$b)\hspace{.1em}\frac{\text{sec}^2 θ}{\text{sin}^2 θ}=\frac{1 + \text{cot}^2 θ}{\text{cos}^2 θ}$$
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#2:
Instructions: verify each identity.
$$a)\hspace{.1em}1 + \text{tan}^2 x=\text{sec}^3 x \hspace{.1em}\text{cos}x$$
$$b)\hspace{.1em}\frac{\text{csc}^2 x}{\text{cos}^2 x}=\text{sec}^2 x + \text{csc}^2 x$$
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#3:
Instructions: verify each identity.
$$a)\hspace{.1em}\text{tan}^2 θ + 1=\text{tan}θ \hspace{.1em}\text{sec}θ \hspace{.1em}\text{csc}θ$$
$$b)\hspace{.1em}\text{csc}^2 θ - 1=\text{csc}^2 θ \hspace{.1em}\text{cos}^2 θ$$
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#4:
Instructions: verify each identity.
$$a)\hspace{.1em}\text{cos}^2 \hspace{.1em}x=\frac{\text{cot}\hspace{.1em}x}{\text{cot}\hspace{.1em}x + \text{tan}\hspace{.1em}x}$$
$$b)\hspace{.1em}\frac{\text{sin}\hspace{.1em}x}{\text{csc}\hspace{.1em}x}=\frac{\text{tan}\hspace{.1em}x}{\text{tan}\hspace{.1em}x + \text{cot}\hspace{.1em}x}$$
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#5:
Instructions: verify each identity.
$$a)\hspace{.1em}\frac{1}{\text{sin}\hspace{.1em}θ}=\text{sin}\hspace{.1em}θ (\text{cot}^2 θ + 1)$$
$$b) \hspace{.1em}\frac{\text{cot}^2 θ}{1 + \text{csc}\hspace{.1em}θ}=\frac{1 - \text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}\text{cot}^2 θ + 1=\text{sec}^2 θ \hspace{.1em}\text{cot}^2 θ$$ $$\hspace{3em}=\frac{1}{\text{cos}^2 θ}\cdot \frac{\text{cos}^2 θ}{\text{sin}^2 θ}$$ $$\hspace{3em}=\frac{1}{\text{sin}^2 θ}$$ $$\hspace{3em}=\text{csc}^2 θ$$ $$\hspace{3em}=\text{cot}^2 θ + 1$$
$$b)\hspace{.1em}\frac{\text{sec}^2 θ}{\text{sin}^2 θ}=\frac{1 + \text{cot}^2 θ}{\text{cos}^2 θ}$$ $$\hspace{3em}=\frac{\text{csc}^2 θ}{\text{cos}^2 θ}$$ $$\hspace{3em}=\Large{\frac{\frac{1}{\text{sin}^2 θ}}{\frac{1}{\text{sec}^2 θ}}}$$ $$\hspace{3em}=\frac{\text{sec}^2 θ}{\text{sin}^2 θ}$$
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#2:
Solutions:
$$a)\hspace{.1em}1 + \text{tan}^2 x=\text{sec}^3 x \hspace{.1em}\text{cos}x$$ $$ \hspace{3em}=\text{sec}^2 x \hspace{.1em}\text{sec}\hspace{.1em}x \text{cos}x$$ $$ \hspace{3em}=(\text{tan}^2 x + 1) \text{sec}\hspace{.1em}x \text{cos}x$$ $$ \hspace{3em}=(\text{tan}^2 x + 1) \frac{1}{\text{cos}\hspace{.1em}x}\text{cos}\hspace{.1em}x$$ $$ \hspace{3em}=1 + \text{tan}^2 x$$
$$b)\hspace{.1em}\frac{\text{csc}^2 x}{\text{cos}^2 x}=\text{sec}^2 x + \text{csc}^2 x$$ $$\hspace{3em}=\frac{1}{\text{cos}^2 x}+ \frac{1}{\text{sin}^2 x}$$ $$\hspace{3em}=\frac{\text{sin}^2 x + \text{cos}^2 x}{\text{sin}^2 x \hspace{.1em}\text{cos}^2 x}$$ $$\hspace{3em}=\frac{1}{\text{sin}^2 x \hspace{.1em}\text{cos}^2 x}$$ $$\hspace{3em}=\frac{\text{csc}^2 x}{\text{cos}^2 x}$$
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#3:
Solutions:
$$a)\hspace{.1em}\text{tan}^2 θ + 1=\text{tan}θ \hspace{.1em}\text{sec}θ \hspace{.1em}\text{csc}θ$$ $$\hspace{3em}=\frac{\text{sin}\hspace{.1em}θ}{\text{cos}\hspace{.1em}θ}\cdot \frac{1}{\text{cos}\hspace{.1em}θ}\cdot \frac{1}{\text{sin}\hspace{.1em}θ}$$ $$\hspace{3em}=\frac{\text{1}}{\text{cos}^2 θ}$$ $$\hspace{3em}=\text{sec}^2 θ$$ $$\hspace{3em}=\text{tan}^2 θ + 1$$
$$b)\hspace{.1em}\text{csc}^2 θ - 1=\text{csc}^2 θ \hspace{.1em}\text{cos}^2 θ$$ $$\hspace{3em}=\frac{\text{cos}^2 θ}{\text{sin}^2 θ}$$ $$\hspace{3em}=\text{cot}^2 θ$$ $$\hspace{3em}=\text{csc}^2 θ - 1$$
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#4:
Solutions:
$$a)\hspace{.1em}\text{cos}^2 \hspace{.1em}x=\frac{\text{cot}\hspace{.1em}x}{\text{cot}\hspace{.1em}x + \text{tan}\hspace{.1em}x}$$ $$ \hspace{3em}=\Large{\frac{\frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}}{\frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}+ \frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}}}$$ $$ \hspace{3em}=\Large{\frac{\frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}}{\frac{\text{cos}^2 x + \text{sin}^2 x}{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}}}$$ $$\hspace{3em}=\Large{\frac{\frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}}{\frac{1}{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}}}$$ $$\hspace{3em}=\frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}\cdot \frac{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}{1}$$ $$\hspace{3em}=\text{cos}^2 \hspace{.1em}x$$
$$b)\hspace{.1em}\frac{\text{sin}\hspace{.1em}x}{\text{csc}\hspace{.1em}x}=\frac{\text{tan}\hspace{.1em}x}{\text{tan}\hspace{.1em}x + \text{cot}\hspace{.1em}x}$$ $$\hspace{3em}=\Large{\frac{\frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}}{\frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}+ \frac{\text{cos}\hspace{.1em}x}{\text{sin}\hspace{.1em}x}}}$$ $$\hspace{3em}=\Large{\frac{\frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}}{\frac{\text{sin}^2 x + \text{cos}^2 x}{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}}}$$ $$\hspace{3em}=\Large{\frac{\frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}}{\frac{1}{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}}}$$ $$\hspace{3em}=\frac{\text{sin}\hspace{.1em}x}{\text{cos}\hspace{.1em}x}\cdot \frac{\text{sin}\hspace{.1em}x \hspace{.1em}\text{cos}\hspace{.1em}x}{1}$$ $$\hspace{3em}=\text{sin}^2 x$$ $$\hspace{3em}=\text{sin}x \hspace{.1em}\text{sin}x$$ $$\hspace{3em}=\frac{\text{sin}\hspace{.1em}x}{\text{csc}\hspace{.1em}x}$$
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#5:
Solutions:
$$a)\hspace{.1em}\frac{1}{\text{sin}\hspace{.1em}θ}=\text{sin}\hspace{.1em}θ (\text{cot}^2 θ + 1)$$ $$\hspace{3em}=\text{sin}\hspace{.1em}θ \hspace{.1em}\text{csc}^2 θ$$ $$\hspace{3em}=\frac{\text{sin}\hspace{.1em}θ}{\text{sin}^2 θ}$$ $$\hspace{3em}=\frac{1}{\text{sin}\hspace{.1em}θ}$$
$$b)\hspace{.1em}\frac{1 - \text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}=\frac{\text{cot}^2 θ}{1 + \text{csc}\hspace{.1em}θ}$$ $$\hspace{3em}=\frac{\text{csc}^2 θ - 1}{1 + \text{csc}\hspace{.1em}θ}$$ $$\hspace{3em}=\frac{(\text{csc}\hspace{.1em}θ + 1)(\text{csc}\hspace{.1em}θ - 1)}{(1 + \text{csc}\hspace{.1em}θ)}$$ $$\frac{1 - \text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}=\text{csc}\hspace{.1em}θ - 1$$ $$\text{csc}\hspace{.1em}θ - 1=\frac{1 - \text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}$$ $$\hspace{3em}=\frac{1}{\text{sin}\hspace{.1em}θ}- \frac{\text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}$$ $$\hspace{3em}=\frac{1}{\text{sin}\hspace{.1em}θ}- 1$$ $$\hspace{3em}=\text{csc}\hspace{.1em}θ - 1$$ $$\frac{1 - \text{sin}\hspace{.1em}θ}{\text{sin}\hspace{.1em}θ}=\text{csc}\hspace{.1em}θ - 1=\frac{\text{cot}^2 θ}{1 + \text{csc}\hspace{.1em}θ}$$
