Question

If the value of \[\sec \theta =\dfrac{5}{4}\] , find the value of \[\dfrac{\sin \theta -2\cos \theta }{\tan \theta -\cot \theta }\] .

Answer 1

Hint: In the given expression, we don’t have any \[\sec \theta \] term. For finding the value of the given expression, we have to find the values of \[\sin \theta \] , \[\cos \theta \] , \[\tan \theta \] , and \[\cot \theta \] . Value of \[\cos \theta \] can be calculated by using the value of \[\sec \theta \]. Putting the value of \[\cos \theta \] in the identity, \[{{\sin }^{2}}\theta +{{\cos }^{2}}=1\], \[\sin \theta \] can be calculated. Then, using the value of \[\cos \theta \] and \[\sin \theta \] , \[\tan \theta \] and \[\cot \theta \] can be calculated. Now, put the values of \[\sin \theta \] , \[\cos \theta \] , \[\tan \theta \] , and \[\cot \theta \] in the given expression and solve it further.

Complete step-by-step answer:

According to the question, it is given that \[\sec \theta =\dfrac{5}{4}\]………………(1)

We know the relation between \[\sec \theta \] and \[\cos \theta \] .

\[\sec \theta =\dfrac{1}{\cos \theta }\]

\[\Rightarrow \cos \theta =\dfrac{1}{\sec \theta }\]……………(2)

Putting the value of \[\sec \theta \] in equation (2), we get

\[ \cos \theta =\dfrac{1}{\dfrac{5}{4}} \]

 \[ \Rightarrow \cos \theta =\dfrac{4}{5} \]

We also have the identity, \[{{\sin }^{2}}\theta +{{\cos }^{2}}=1\] .

Taking \[{{\cos }^{2}}\theta \] to RHS, we get \[{{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \]………………(3)

Putting the value of \[\cos \theta \] in equation (3), we get

\[ {{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \]

 \[ \Rightarrow {{\sin }^{2}}\theta =1-\dfrac{16}{25} \]

 \[ \Rightarrow {{\sin }^{2}}\theta =\dfrac{25-16}{25} \]

\[\Rightarrow {{\sin }^{2}}\theta =\dfrac{9}{25}\]…………….(4)

Taking root in both LHS and RHS of the equation (4), we get

\[\sin \theta =\dfrac{3}{5}\]…………………….(5)

We also know that, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\]………………(6)

Using the values of \[\sin \theta \] and \[\cos \theta \] in equation (6), we get

\[\tan \theta =\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}\]

\[\Rightarrow \tan \theta =\dfrac{3}{4}\] ………….(7)

We also know the relation between \[\tan \theta \] and \[\cot \theta \],

\[\cot \theta =\dfrac{1}{\tan \theta }\]

\[\Rightarrow \cot \theta =\dfrac{1}{\dfrac{3}{4}}\]

\[\Rightarrow \cot \theta =\dfrac{4}{3}\]………..(8)

Now, we have the values of \[\sin \theta \] , \[\cos \theta \] , \[\tan \theta \] , and \[\cot \theta \] .

Putting these values in the expression, \[\dfrac{\sin \theta -2\cos \theta }{\tan \theta -\cot \theta }\] we get,

\[ \dfrac{\sin \theta -2\cos \theta }{\tan \theta -\cot \theta } \]

 \[ =\dfrac{\dfrac{3}{5}-2.\dfrac{4}{5}}{\dfrac{3}{4}-\dfrac{4}{3}} \]

 \[ =\dfrac{\dfrac{3-8}{5}}{\dfrac{9-16}{12}} \]

 \[ =\dfrac{\dfrac{-5}{5}}{\dfrac{-7}{12}} \]

 \[ =\dfrac{-1}{\dfrac{-7}{12}} \]

 \[ =\dfrac{12}{7} \]

Therefore, \[\dfrac{\sin \theta -2\cos \theta }{\tan \theta -\cot \theta }=\dfrac{12}{7}\] .

Note: This question can also be solved using Pythagoras theorem.

\[\sec \theta =\dfrac{5}{4}\]

Using Pythagoras theorem, we can find height.

Height= \[\sqrt{{{\left( hypotenuse \right)}^{2}}-{{\left( base \right)}^{2}}}\]

\[ =\sqrt{{{\left( 5 \right)}^{2}}-{{\left( 4 \right)}^{2}}} \]

 \[ =\sqrt{25-16} \]

 \[ =\sqrt{9} \]

 \[ =3 \]

\[ \sin \theta =\dfrac{height}{hypotenuse} \]

 \[ \sin \theta =\dfrac{3}{5} \]

\[ \cos \theta =\dfrac{base}{hypotenuse} \]

\[ \cos \theta =\dfrac{4}{5} \]

\[ \tan \theta =\dfrac{height}{base} \]

\[ \tan \theta =\dfrac{3}{4} \]

\[ \cot \theta =\dfrac{base}{height} \]

 \[ \cot \theta =\dfrac{4}{3} \]

Now, we have the values of \[\sin \theta \] , \[\cos \theta \] , \[\tan \theta \] , and \[\cot \theta \] .Put these values in the expression, \[\dfrac{\sin \theta -2\cos \theta }{\tan \theta -\cot \theta }\] and solve it further.