Question

simplify . sin ( 180° - theta ) / tan ( 90° + theta) × cos ( 360° - theta ) / tan ( 180° + theta ) × cot ( 90° - theta ) /. sin (- theta)​

Answer 1

Given :-

$\sf \dfrac{sin(180^{\circ}-\theta)}{tan(90^{\circ}+\theta)} \times \dfrac{cos(360^{\circ}-\theta)}{tan(180^{\circ}+\theta)}\times\dfrac{cot(90^{\circ}-\theta)}{sin(-\theta)}$

Concept to know :-

We know that the trigonometric functions of certain angles in terms of angle in Quadrant are

$\sf sin(180^{\circ}- \theta) = +sin\theta$

$\sf tan(90^{\circ}+\theta) = -cot\theta$

$\sf cos(360^{\circ} - \theta) = +cos\theta$

$\sf tan(180^{\circ} +\theta) = +tan\theta$

$\sf cot(90^{\circ} -\theta) = +tan\theta$

Also we know that ,

$\sf sin(-\theta) = sin\theta$

Using these formula We get ,

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$= \sf \dfrac{sin\theta}{-cot\theta} \times \dfrac{+cos\theta}{+tan\theta}\times \dfrac{+tan\theta}{-sin\theta}$

$= \sf \dfrac{\not sin\theta}{cot\theta} \times \dfrac{+cos\theta}{\not +tan\theta}\times \dfrac{\not +tan\theta}{\not sin\theta}$

$=\sf \dfrac{cos\theta}{cot\theta}$

$=\sf \dfrac{cos\theta}{\dfrac{cos\theta}{sin\theta} }$

$\sf =\dfrac{(cos\theta )(sin\theta)}{cos\theta}$

$\sf = sin\theta$

So, the required answer is$\sf = sin\theta$

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Know more :-

• sin(-A) = -sinA
• cos(-A) = cosA
• tan(-A) = -tanA
• csc(-A) = -cscA
• sec(-A) = secA
• cot(-A) = -cotA

In Quadrant - 1 :- All ratios are positive

In Quadrant- 2 :- sin, csc are positive

In Quadrant- 3 :- tan , cot are positive

In Quadrant-4 :- sec, cos are positive

Answer 2

Step-by-step explanation:

hope this helps you study well!!