Choose the correct option. Justify your choice.
(i) $9 \sec ^{2} \mathrm{~A}-9 \tan ^{2} \mathrm{~A}=$
(A) $\underline{1}$
(B) $\underline{9}$
(C) 8
(D) 0
(ii) $(1+\tan \theta+\sec \theta)(1+\cot \theta-\operatorname{cosec} \theta)=$
(A) 0
(B) 1
(C) 2
(D) $-1$
(iii) $(\sec A+\tan A)(1-\sin A)=$
(A) $\sec A$
(B) $\sin \mathrm{A}$
(C) $\operatorname{cosec} \mathrm{A}$
(D) $\cos \mathrm{A}$
(iv) $\frac{1+\tan ^{2} A}{1+\cot ^{2} A}=$
(A) $\sec ^{2} A$
(B) $-1$
(C) $\cot ^{2} \mathrm{~A}$
(D) $\tan ^{2} \mathrm{~A}$

Answer

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Precalculus

NCERT Class 10 - Math

Chapter 8

Introduction to Trigonometry

Section 4

Trigonometric Ratios of Complementary Angles

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Video Transcript

So in the given question we have a few trigonometry equation equations, expressions to be evaluated the first one has given us 96 square a minus my entrance square. A. So in order to evaluate this, we can use the identity that six square three to minus trans squared theta is equal to one. So nine times six square a minus trance square A. Yes equal to nine times 1 which is equal to nine. Now the next expression which is given is one plus 10 ft up plus sick data times one plus Cartier to minus Kosik tita so we can expand this expression and we would get one plus card data minus Kosik data. Place we have Panty to times Cartier to my understand peter crimes Kosik peter plus C. T. To sick peter plus six theatre times got theater minus six peter times Cosine theta. This is what we get. We have a tan theta also over here and now we should know that Partita is equal to one by tan Theta and then Panty to Times card data would then be well to one right? And similarly if we take Dante to as scientific to divided by cost data and capacity to as one by saying data 2030 times Kosik theta is equal to one by cost data. And since six theta is equal to one by costly to We can write 22 times music theater, the Sikh Theatre and next what we have is similarly we can write with court data which is costly to divided by sine theta. So called theta times sick theater would then be equal to one by sine theta which is classic theater. So this is Kosik painter and then what we have is we have why 6ft our here. So we have minus sick pita and plastic leader which would become zero. Then we have plus cost a to minus cost data which will also become zero. And what is left is one plus Cartier to plus anti to plus one minus sick peter Kosik data. Right so now we can simplify this expression. So one plus Cartier to Pakistan data. This constant er divided by scientists to plus scientists to divided by cost data Plus one. So we can turn this once and right at us too. Two minus city to city to can be redressed one bite cost data times. Scientific. So now what we have is we can cross multiply these two terms. So we will have to place cost square teeter plus sine square theater divided by cross the town scientist to and since we have the identity that cost where to. Topless science Square theater is equal to one. We can write this expression would simplify too two plus two plus one by pasta to scientific data minus one by cost data scientist to It's just equal to two. So the required value of the second explosion that has given the question is equal to two. Now let's move on to the third expression that is given in the question. So the third expression is sick A plus an A Times 1- Sign in. So here we can do the same. We can use the same identities to write seek a blast on AS1 by cost a plus sign. A divided by costing times 19 is saying and this would then be equal to 1- Sign. A one minus sine eight times one plus sign which is one square minus sine squared egg noticed one minus sine square egg divided by cost of A. And we know the identity cause square a square. The topless Science Square theater is equal to one from which we can right? Because a cost square A is equal to one minus sine squared eight. So over here we would have cost square A divided by cost A. Which is equal to the cost of A. So this is what we get after simplifying the given expression. And lastly we have 1/4 expression that has given us one plus times square a. Divided by one Place Court Square A. So we have do I rented this that have these terms which are one plus and square Theta is equal to six square theater and one plus court Square theater is equal to go six square theater. Then this would be equal to six square feet to divided by CO. Six square tita sick sick peter is run by core square peter C one by cost data. So we have 616 square feet as one by cost square theater and CO. Six square theatre as one by Science Square Theater, which would be equal to sine square feet are divided by cause square theater. And this would give us scientific delivered by cal state a whole square, which is actually tan squared theta, right? So one plus stance. We're ready. Worried by one plus court square, they would simplify took the expression tan square in. So this is the required answers for the human expressions. I hope you understood the answer. Thank you.

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