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(a) The function is $\"\\tan$ .

Differentiate on each side with respect to $\"x\"$ .

$\"\\frac{d}{dx}(\\tan$ .

Derivative of the tangent function: $\"\\frac{d}{dx}(\\tan$ .

Quotient rule of derivatives: $\"\\left$ .

$\"\\sec^2$

Derivative of the cosine function: $\"\\frac{d}{dx}(\\cos$ .

Derivative of the sine function: $\"\\frac{d}{dx}(\\sin$ .

$\"\\sec^2$ .

$\"\\sec^2$

Pythagorean identity: $\"\\cos^2$ .

$\"\\sec^2$ .

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(b) $\"\\sec$ .

Differentiate on each side with respect to $\"x\"$ .

$\"\\frac{d}{dx}(\\sec$

Derivative of the secant function: $\"\\frac{d}{dx}(\\sec$ .

Quotient rule of derivatives: $\"\\left$ .

$\"\\\\(\\sec$

$\"\\\\\\sec$ .

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(c) $\"\\\\\\sin$ .

Differentiate on each side with respect to $\"x\"$ .

$\"\\\\\\frac{d}{dx}(\\sin$

$\"(\\cos$

Derivative of the cotangent function: $\"\\frac{d}{dx}(\\cot$ .

Derivative of the cosecant function: $\"\\frac{d}{dx}(\\csc$ .

$\"(\\cos$

$\"\\\\(\\cos$

Pythagorean identity: $\"\\csc^2$ .

$\"\\\\=\\left$

$\"\\\\(\\cos$ .

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(a) $\"\\sec^2$ .

(b) $\"\\\\\\sec$ .

(c) $\"\\\\(\\cos$ .