Question

If the value of the trigonometric ratio, i.e. ${\text{cot}}\theta {\text{ = }}\dfrac{{\text{7}}}{8}$ then find the value of $\dfrac{{\left( {{\text{1 + sin}}\theta } \right)\left( {{\text{1 - sin}}\theta } \right)}}{{\left( {{\text{1 + cos}}\theta } \right)\left( {1 - {\text{cos}}\theta } \right)}}$

Answer 1

Hint: In this question, we have been given the value of cot function, using that and a basic trigonometric identity which includes sine and cosine function, we simplify the equation into known trigonometric identity. Then whenever it is required we will use the following formula. i.e.
${\text{si}}{{\text{n}}^2}\theta {\text{ + co}}{{\text{s}}^2}\theta = 1 \\
    \\
$

Complete step-by-step answer:
Given data
${\text{cot}}\theta {\text{ = }}\dfrac{{\text{7}}}{8}$
Now $\dfrac{{\left( {{\text{1 + sin}}\theta } \right)\left( {{\text{1 - sin}}\theta } \right)}}{{\left( {{\text{1 + cos}}\theta } \right)\left( {1 - {\text{cos}}\theta } \right)}}$= $\dfrac{{{\text{1 + sin}}\theta {\text{ - sin}}\theta {\text{ - si}}{{\text{n}}^2}\theta }}{{{\text{1 + cos}}\theta {\text{ - cos}}\theta {\text{ - co}}{{\text{s}}^2}\theta }}$
$ \Rightarrow \dfrac{{{\text{1 - si}}{{\text{n}}^2}\theta }}{{{\text{1 - co}}{{\text{s}}^2}\theta }}{\text{ - Equation 1}}$

As we know $
  {\text{si}}{{\text{n}}^2}\theta {\text{ + co}}{{\text{s}}^2}\theta = 1 \\
    \\
$
$
   \Rightarrow {\text{si}}{{\text{n}}^2}\theta = 1 - {\text{co}}{{\text{s}}^2}\theta {\text{ and}} \\
  {\text{ }}{\cos ^2}\theta = 1 - {\text{si}}{{\text{n}}^2}\theta \\
$

Using this relation to solve Equation 1 we get
$
   \Rightarrow \dfrac{{{\text{1 - si}}{{\text{n}}^2}\theta }}{{{\text{1 - co}}{{\text{s}}^2}\theta }} = \dfrac{{{\text{co}}{{\text{s}}^2}\theta }}{{{\text{si}}{{\text{n}}^2}\theta }}. \\
   \Rightarrow {\text{co}}{{\text{t}}^2}\theta {\text{ }}\left( {{\text{as cot}}\theta {\text{ = }}\dfrac{{{\text{cos}}\theta }}{{{\text{sin}}\theta }}} \right) \\
$
Therefore, $\dfrac{{\left( {{\text{1 + sin}}\theta } \right)\left( {{\text{1 - sin}}\theta } \right)}}{{\left( {{\text{1 + cos}}\theta } \right)\left( {1 - {\text{cos}}\theta } \right)}}$=${\left( {\dfrac{7}{8}} \right)^2} = \dfrac{{49}}{{64}}$.

Note: In such types of questions analyze the equations and perform basic mathematical operations. Then use trigonometric identities such that the required part of the problem can be reduced into a known or given trigonometric ratio. Trigonometric identities come in very handy for tackling this kind of problem.