10th Class Mathematics Introduction to Trigonometry Question Bank MCQs - Introduction to Trigonometry

If \[{{\tan }^{2}}\theta +{{\cot }^{2}}\theta =2,\] \[\theta \]is an acute angle, then \[{{\tan }^{3}}\theta +{{\cot }^{3}}\theta \] is equal to:

A) 2

B) 3

C) 4

D) 8

Correct Answer: A

Solution :

[a] \[{{\tan }^{2}}\theta +{{\cot }^{2}}\theta =2\]

\[\Rightarrow \,\,\,{{\tan }^{2}}\theta +{{\cot }^{2}}\theta =2\tan \theta \cdot \cot \theta \]

\[\Rightarrow \,\,\,\,\,\,\,\,\,{{(\tan \theta -\cot \theta )}^{2}}=0\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\tan \theta -\cot \theta =0\Rightarrow \,\tan \theta =\cot \theta \]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\theta =45{}^\circ \]

\[\therefore \,\,\,\,\,{{\tan }^{3}}\theta +{{\cot }^{3}}\theta ={{\tan }^{3}}45{}^\circ +{{\cot }^{3}}45{}^\circ ={{(1)}^{3}}+{{(1)}^{3}}=2\]