Question
3) Simpilfy: \( \boldsymbol{\operatorname { s i n }} \theta(\csc \theta-\sin \theta)= \) a) \( \cos ^{2} \theta \) b) \( \sin ^{2} \theta \) c) \( \cot ^{2} \theta \) d) \( \tan ^{2} \theta \)
Ask by Hampton Rose. in Puerto Rico
Dec 03,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \cos^{2} \theta \).
Therefore, the correct answer is a) \( \cos^{2} \theta \).
Solution
Calculate or simplify the expression \( \sin(\theta)(\csc(\theta)-\sin(\theta)) \).
Simplify the expression by following steps:
- step0: Solution:
\(\sin\left(\theta \right)\left(\csc\left(\theta \right)-\sin\left(\theta \right)\right)\)
- step1: Transform the expression:
\(\sin\left(\theta \right)\times \frac{1-\sin^{2}\left(\theta \right)}{\sin\left(\theta \right)}\)
- step2: Reduce the fraction:
\(1-\sin^{2}\left(\theta \right)\)
- step3: Transform the expression:
\(\cos^{2}\left(\theta \right)\)
The simplified expression is \( \cos^{2}(\theta) \).
Therefore, the correct answer is a) \( \cos^{2} \theta \).
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Simplify this solution The Deep Dive
To simplify the expression \( \sin \theta(\csc \theta - \sin \theta) \), we start by substituting \( \csc \theta \) with \( \frac{1}{\sin \theta} \): \[ \sin \theta \left( \frac{1}{\sin \theta} - \sin \theta \right) = \sin \theta \left( \frac{1 - \sin^2 \theta}{\sin \theta} \right) = 1 - \sin^2 \theta \] Using the Pythagorean identity, \( 1 - \sin^2 \theta = \cos^2 \theta \). Thus, the simplified expression is \( \cos^2 \theta \). The correct answer is: a) \( \cos^2 \theta \)
