# 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

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