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Trigonometric Identities — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsTrigonometric Identities2 Marks
Question
$
\text { If } \mathrm{x}=\mathrm{r} \sin \mathrm{A} \cos \mathrm{B}, \mathrm{y}=\mathrm{r} \sin \mathrm{A} \sin \mathrm{B} \text { and } \mathrm{z}=\mathrm{r} \cos \mathrm{A} \text {, prove that } x^2+y^2+z^2=r^2
$
✓

Answer

we get:
$
\begin{aligned}
& x^2=(a \cos \theta)^2=a^2 \cos ^2 \theta \\
& y^2=(b \cot \theta)^2=b^2 \cot ^2 \theta \\
& \text { LHS }=\frac{a^2}{x^2}-\frac{b^2}{y^2}=\frac{a^2}{a^2 \cos ^2 \theta}-\frac{b^2}{b^2 \cot ^2 \theta}=\frac{1}{\cos ^2 \theta}-\frac{1}{\cot ^2 \theta} \\
& \Rightarrow \text { LHS }=\sec ^2 \theta-\tan ^2 \theta=1\left[\text { Since } 1+\tan ^2 \theta=\sec ^2 \theta\right]
\end{aligned}
$

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