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Introduction to Trigonometry — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsIntroduction to Trigonometry3 Marks
Question
Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
✓

Answer

We have to express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
For $\sin A$, By using identity $cosec ^ { 2 } A - \cot ^ { 2 } A = 1$
$\Rightarrow cosec ^ { 2 } A = 1 + \cot ^ { 2 } A$
$\Rightarrow \frac { 1 } { \sin ^ { 2 } A } = 1 + \cot ^ { 2 } A$
$\Rightarrow \sin ^ { 2 } A = \frac { 1 } { 1 + \cot ^ { 2 } A }$
$ \Rightarrow \quad \sin A = \frac { 1 } { \sqrt { 1 + \cot ^ { 2 } A } }$
 For sec A, By using identity $ \sec ^ { 2 } A - \tan ^ { 2 } A = 1$
$ \Rightarrow \sec ^ { 2 } A = 1 + \tan ^ { 2 } A$
$ \Rightarrow \sec ^ { 2 } A = 1 + \frac { 1 } { \cot ^ { 2 } A } = \frac { \cot ^ { 2 } A + 1 } { \cot ^ { 2 } A }$
$ \Rightarrow \sec ^ { 2 } A = \frac { 1 + \cot ^ { 2 } A } { \cot ^ { 2 } A }$
$ \Rightarrow \sec A = \frac { \sqrt { 1 + \cot ^ { 2 } A } } { \cot A }$ For tanA, $ \tan A = \frac { 1 } { \cot A }$

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