Question

If $\theta$ is in the first quadrant and cos $\theta =\frac{3}{5}$, then the value of $\frac{5tan\theta -4cosec\theta }{5sec\theta -4cot\theta }$ is

• A. $\frac{5}{34}$
• B. $\frac{5}{16}$
• C. $\frac{5}{-34}$
• D. $\frac{-5}{16}$

Answer 1

The correct option is B $\frac{5}{16}$

$\theta isinthefirstquardrant,soalltrigonometricalfunctionswillhavepositivevalues.Givencos\theta =\frac{3}{5}\therefore sin\theta =\sqrt{1-{cos}^2\theta }=\frac{4}{5}\therefore tan\theta =\frac{sin\theta }{cos\theta }=\frac{4}{3}\therefore cosec\theta =\frac{1}{sin\theta }=\frac{5}{4}\therefore sec\theta =\frac{1}{cos\theta }=\frac{5}{3}andcot\theta =\frac{1}{tan\theta }=\frac{3}{4}substitutingthesevaluesofthetrigonometricfunctionsin\frac{5tan\theta -4cosec\theta }{5sec\theta -4cot\theta }itbecomes\frac{5\times \frac{4}{3}-4\times \frac{5}{4}}{5\times \frac{5}{3}-4\times \frac{3}{4}}=\frac{5}{16}$