# 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) $(1 + tan\ θ + sec\ θ) (1 + cot\ θ - \operatorname{cosec} θ ) =$(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 (D) $cos\ A$(iv) $\frac{1+tan^2\ A}{1+cot^2\ A}$(A) $sec^2\ A$ (B) $-1$ (C) $cot^2\ A$ (D) $tan^2\ A$.

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To do:

We have to choose the correct option and justify it.

Solution:

(i) We know that,

$sec^2\ \theta - tan^2\ \theta =1$

Therefore,

$9\ sec^2\ A - 9\ tan^2 A = 9(sec^2\ A - tan^2 A)$

$=9(1)$

$=9$

The correct option is B.

(ii)$(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=(\frac{1}{1}+\frac{\sin \theta}{\cos \theta}+\frac{1}{\cos \theta})(\frac{1}{1}+\frac{\cos \theta}{\sin \theta}-\frac{1}{\sin \theta})$

$=(\frac{\cos \theta+\sin \theta+1}{\cos \theta})(\frac{\sin \theta+\cos \theta-1}{\sin \theta})$

$=\frac{(\cos \theta+\sin \theta)^{2}-(1)^{2}}{\cos \theta \sin \theta}$

$=\frac{\cos ^{2} \theta+\sin ^{2} \theta+2 \cos \theta \sin \theta-1}{\cos \theta \sin \theta}$

$=\frac{1+2 \cos \theta \sin \theta-1}{\cos \theta \sin \theta}$

$=\frac{2 \cos \theta \sin \theta}{\cos \theta \sin \theta}$

$=2$

The correct option is $C$.

(iii) $(\sec \mathrm{A}+\tan \mathrm{A})(1-\sin \mathrm{A})=(\frac{1}{\cos \mathrm{A}}+\frac{\sin \mathrm{A}}{\cos \mathrm{A}})(\frac{1-\sin \mathrm{A}}{1})$

$=(\frac{1+\sin \mathrm{A}}{\cos \mathrm{A}})(\frac{1-\sin \mathrm{A}}{1})$

$=\frac{(1)^{2}-(\sin \mathrm{A})^{2}}{\cos \mathrm{A}}$

$=\frac{1-\sin ^{2} \mathrm{~A}}{\cos \mathrm{A}}$

$=\frac{\cos ^{2} \mathrm{~A}}{\cos \mathrm{A}}$

$=\cos \mathrm{A}$

The correct option is D.

(iv) $\frac{1+\tan ^{2} \mathrm{~A}}{1+\cot ^{2} \mathrm{~A}}=\frac{\sec ^{2} \mathrm{~A}}{\operatorname{cosec}^{2} \mathrm{~A}}$

$=\frac{\frac{1}{\cos ^{2} \mathrm{~A}}}{\frac{1}{\sin ^{2} \mathrm{~A}}}$

$=\frac{1}{\cos ^{2} \mathrm{~A}} \times \frac{\sin ^{2} \mathrm{~A}}{1}$

$=\frac{\sin ^{2} \mathrm{~A}}{\cos ^{2} \mathrm{~A}}$

$=\tan ^{2} \mathrm{~A}$

The correct option is D.

Updated on 10-Oct-2022 13:22:36