$
{\text{If sin}}\theta {\text{, cos}}\theta {\text{ and tan}}\theta {\text{ are in G}}{\text{.P}}{\text{. then co}}{{\text{t}}^6}\theta - {\cot ^2}\theta {\text{ is}} \\
{\text{A}}{\text{. 1}} \\
{\text{B}}{\text{. }}\dfrac{1}{2} \\
{\text{C}}{\text{. 2}} \\
{\text{D}}{\text{. 3}} \\
$
Last updated date: 25th Mar 2023
•
Total views: 312.9k
•
Views today: 8.92k
Answer
312.9k+ views
$
\\
{\text{We know that when }}a,{\text{ }}b,{\text{ }}c{\text{ are in GP then }} \\
\Rightarrow {b^2} = a \cdot c \\
{\text{here sin}}\theta {\text{, cos}}\theta {\text{ and tan}}\theta {\text{ are in gp}} \\
\Rightarrow {\text{co}}{{\text{s}}^2}\theta = \sin \theta \cdot \tan \theta \\
{\text{ = sin}}\theta \cdot \dfrac{{\sin \theta }}{{\cos \theta }} \\
\Rightarrow \dfrac{{{\text{co}}{{\text{s}}^2}\theta }}{{{\text{si}}{{\text{n}}^2}\theta }} = \dfrac{1}{{\cos \theta }}{\text{ }} \Rightarrow \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^3}\theta }} = 1{\text{ }}........{\text{(i)}} \\
\Rightarrow {\cot ^2}\theta = \sec \theta \\
{\text{Now put the value of }}{\cot ^2}\theta {\text{ in the question}} \\
\Rightarrow {\cot ^6}\theta - {\cot ^2}\theta = {\sec ^3}\theta - \sec \theta \\
{\text{ = }}\sec \theta ({\sec ^2}\theta - 1) \\
{\text{ = }}\sec \theta \cdot {\tan ^2}\theta \\
{\text{ = }}\dfrac{1}{{\cos \theta }} \cdot \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }} \\
{\text{ = }}\dfrac{{{{\sin }^2}\theta }}{{{{\cos }^3}\theta }} \\
{\text{By putting the value in equation (i)}} \\
{\cot ^6}\theta - {\cot ^2}\theta = 1{\text{ }} \\
{\text{So option A is correct}}{\text{.}} \\
{\text{Note: - Always try to use geometric mean when three consecutive term of a GP are given}}{\text{. }} \\
{\text{these are the best method to solve the questions}}{\text{.}} \\
$
\\
{\text{We know that when }}a,{\text{ }}b,{\text{ }}c{\text{ are in GP then }} \\
\Rightarrow {b^2} = a \cdot c \\
{\text{here sin}}\theta {\text{, cos}}\theta {\text{ and tan}}\theta {\text{ are in gp}} \\
\Rightarrow {\text{co}}{{\text{s}}^2}\theta = \sin \theta \cdot \tan \theta \\
{\text{ = sin}}\theta \cdot \dfrac{{\sin \theta }}{{\cos \theta }} \\
\Rightarrow \dfrac{{{\text{co}}{{\text{s}}^2}\theta }}{{{\text{si}}{{\text{n}}^2}\theta }} = \dfrac{1}{{\cos \theta }}{\text{ }} \Rightarrow \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^3}\theta }} = 1{\text{ }}........{\text{(i)}} \\
\Rightarrow {\cot ^2}\theta = \sec \theta \\
{\text{Now put the value of }}{\cot ^2}\theta {\text{ in the question}} \\
\Rightarrow {\cot ^6}\theta - {\cot ^2}\theta = {\sec ^3}\theta - \sec \theta \\
{\text{ = }}\sec \theta ({\sec ^2}\theta - 1) \\
{\text{ = }}\sec \theta \cdot {\tan ^2}\theta \\
{\text{ = }}\dfrac{1}{{\cos \theta }} \cdot \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }} \\
{\text{ = }}\dfrac{{{{\sin }^2}\theta }}{{{{\cos }^3}\theta }} \\
{\text{By putting the value in equation (i)}} \\
{\cot ^6}\theta - {\cot ^2}\theta = 1{\text{ }} \\
{\text{So option A is correct}}{\text{.}} \\
{\text{Note: - Always try to use geometric mean when three consecutive term of a GP are given}}{\text{. }} \\
{\text{these are the best method to solve the questions}}{\text{.}} \\
$
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
