Choose the correct option. Justify your choice :

Question :

Choose the correct option. Justify your choice :

(i)  \(9 sec^2 A – 9 tan^2 A\) =

(a)  1

(b)  9

(c)  8

(d)  0

(ii)  \((1 – tan \theta + sec \theta)\)\((1 + cot \theta – cosec \theta)\) =

(a)  0

(b)  1

(c)  2

(d) -1

(iii)  (sec A + tan A)(1 – sin A) =

(a)  sec A

(b)  sin A

(c)  cosec A

(c)  cos A

(iv)  \(1 + tan^2 A\over 1 + cot^2 A\) =

(a)  \(sec^2 A\)

(b)  -1

(c)  \(cot^2 A\)

(d)  \(tan^2 A\)

Solution :

(i)  (b)  Because : \(9 sec^2 A – 9 tan^2 A\) = 9(\(9 sec^2 A – 9 tan^2 A\)) =  9(1) = 9

(ii)  (c)  Because : \((1 – tan \theta\) + sec \theta)\)\((1 + cot \theta – cosec \theta)\)

= (1 + \(sin\theta\over cos\theta\) + \(1\over cos\theta\))(1 + \(cos\theta\over sin\theta\) – \(1\over sin\theta\))

= \(({cos\theta +sin\theta})^2 – 1\over cos\theta sin\theta \)

= \(1 + 2cos\theta sin\theta – 1\over sin\theta cos\theta\) = \(2sin\theta cos\theta\over sin\theta cos\theta\) = 2

(iii)  (d)  Because

(sec A + tan A)(1 – sin A) = (\(1\over cos A\) + \(sin A\over cos A\))(1 – sin A) = (\(1 + sin A\over cos A\))(1 – sin A)

= \(1 – sin^2A\over cos A\) = \(cos^2 A\over cos A\) = cos A

(iv)  (d)  Because

\(1 + tan^2 A\over 1 + cot^2 A\) = \(1 + tan^2 A\over 1 + {1\over tan^2 A}\) = (\(1 + tan^2 A\)) \(\times\) \(tan^2 A\over 1 + tan^2 A\) = \(tan^2 A\)

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