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Introduction of Trigonometry and its Application — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsIntroduction of Trigonometry and its Application1 Mark
MCQ
Choose the correct answer from the given four options.
If $\cos(\alpha+\beta)=0,$ then $\sin(\alpha-\beta)$ can be reduced to:
  • A
    $\cos\beta$
  • ✓
    $\cos2\beta$
  • C
    $\sin\alpha$
  • D
    $\sin2\alpha$
✓

Answer

Correct option: B.
$\cos2\beta$
$\text{Given, }\cos(\alpha+\beta)=0=\cos90^\circ$ $[\because\cos90^\circ=0]$
$\Rightarrow\ \alpha+\beta=90^\circ$
$\Rightarrow\ \alpha=90^\circ-\beta\ \ ...(\text{i})$
$\text{Now, }\sin(\alpha-\beta)=\sin(90^\circ-\beta-\beta)$ [put the value from Eq. (i)]
$=\sin(90^\circ-2\beta)$
$=\cos2\beta$ $[\because\sin(90^\circ-\theta)=\cos\theta]$
Hence, $\sin(\alpha-\beta)$ can be reduced to $\cos2\beta.$

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