Question

The expression$\frac{tan\, A}{1-cot\, A}+\frac{cot\, A}{1-tan \, A}$ can be written as

• A. $sin A \,cos A\, +1$
• B. $secA \,cosecA + 1$
• C. $tanA \,+\, cotA$
• D. $secA \,+ \,cosecA$

Answer 1

Answer: (B)

Given expression is
$\frac{tan A}{1-cot A}+\frac{cot A}{1-tan A}$
$=\frac{sin A}{cos A}\times\frac{sin A}{sin A-cos A}+\frac{cos A}{sin A}\times\frac{cos A}{cos A-sin A}$
$=\frac{1}{sin A-cos A}\Big\{\frac{sin^3 A-cos^3 A}{cos A sin A}\Big\}$
$=\frac{sin^2+sin A cos A+cos^2 A}{sin A cos A}$
$=\frac{1+sin A cos A}{sin A cos A}=1+sec A cosec A$