$\"\"$

The function is $\"f\\left$ .

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

$\"f\'\\left$

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

$\"\\\\f\'\\left$

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

Pythagorean identity: $\"\\sec^2x-\\tan$ .

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

$\"\\\\f\'\\left$ .

$\"\"$

(b) $\"f\\left$ .

Quotient identity: $\"\\tan$ and $\"\\sec$ .

Substitute above values in the function.

$\"\\\\f\\left$

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

$\"\\\\\\frac{d}{dx}f\\left$

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

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

$\"\\\\f\'\\left$

$\"\"$

(c)

Consider the result of (a), $\"\\\\f\'\\left$ .

$\"\\\\f\'\\left$ .

Quotient identity: $\"\\tan$ and $\"\\cos$ .

$\"\\\\f\'\\left$

$\"\\\\f\'\\left$ .

Above result is same as the result of (b).

Therefore results of (a) and (b) are same.

$\"\"$

(a) $\"\\\\f\'\\left$ .

(b) $\"\\\\f\'\\left$ .