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theres 2 fill in the blank boxes and 3 drop down menus, below i will list the options in the drop down menus.box 1 - apply quotient identities, apply Pythagorean identities, apply double-number identities, apply even-odd identities.box 2 - apply cofunction identities, use the definition of subtraction, apply even-odd identities, Write as one expresssion combine like terms.box 3 - apply cofunction identities, apply double-number identities, apply Pythagorean identities, apply even-odd identities.

theres 2 fill in the blank boxes and 3 drop down menus below i will list the options in the drop down menusbox 1 apply quotient identities apply Pythagorean ide class=

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Solution

Box 1 : Apply Quotient Identities

[tex]cotx-tanx=\frac{cosx}{sinx}-\frac{sinx}{cosx}[/tex]

The answer for the first box is

[tex]\begin{equation*} \frac{cosx}{sinx}-\frac{sinx}{cosx} \end{equation*}[/tex]

Box 2: Write as one expression

[tex]\begin{gathered} cotx-tanx=\frac{cosx}{s\imaginaryI nx}-\frac{s\imaginaryI nx}{cosx} \\ cotx-tanx=\frac{cosx(cosx)-sinx(sinx)}{sinxcosx} \\ cotx-tanx=\frac{cos^2x-sin^2x}{sinxcosx} \end{gathered}[/tex]

The answer for the second box is

[tex]\frac{cos^{2}x-s\imaginaryI n^{2}x}{s\imaginaryI nxcosx}[/tex]

Before the box 3, please note the identity

Note: Trigonometry I dentities

[tex]\begin{gathered} cos^2x-s\mathrm{i}n^2x=cos2x \\ 2sinxcosx=sin2x \end{gathered}[/tex]

Box 3: Apply Double - Number Identities

[tex]\begin{gathered} cotx-tanx=\frac{cos^{2}x-s\imaginaryI n^{2}x}{s\imaginaryI nxcosx} \\ Applying\text{ the above trigonometry identities} \\ cotx-tanx=\frac{cos2x}{sinxcosx} \\ cotx-tanx=\frac{cos2x}{sinxcosx}\times\frac{2}{2} \\ cotx-tanx=\frac{2cos2x}{2sinxcosx} \\ cotx-tanx=\frac{2cos2x}{sin2x} \end{gathered}[/tex]

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