In a right-angled triangle, one acute angle \(A\) satisfies \(\cos A=\displaystyle\frac{20}{29}\). What is \(\mathrm{cosec}\,A+\cot A\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{7}{3}\)
\(\cos A\) is adjacent side ÷ hypotenuse. With the adjacent side 20 and the hypotenuse 29, Pythagoras gives the opposite side; \(A\) is acute, so all ratios are positive.
Opposite side \(=\sqrt{29^2-20^2}=\sqrt{841-400}=\sqrt{441}=21\)
\(\mathrm{cosec}\,A=\displaystyle\frac{29}{21}\) and \(\cot A=\displaystyle\frac{20}{21}\)
\(\mathrm{cosec}\,A+\cot A=\displaystyle\frac{29+20}{21}=\frac{49}{21}=\frac{7}{3}\)