Choose the correct option and justify your choice :
(i) $\frac{2 \tan 30^{\circ}}{1+\tan ^{2} 30^{\circ}}=$
(A) $\sin 60^{\circ}$
(B) $\cos 60^{\circ}$
(C) $\tan 60^{\circ}$
(D) $\sin 30^{\circ}$
(ii) $\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}=$
(A) $\tan 90^{\circ}$
(B) 1
(C) $\sin 45^{\circ}$
(D) 0
(iii) $\sin 2 \mathrm{~A}=2 \sin \mathrm{A}$ is true when $\mathrm{A}=$
(A) $0^{\circ}$
(B) $30^{\circ}$
(C) $45^{\circ}$
(D) $60^{\circ}$
(iv) $\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}=$
(A) $\cos 60^{\circ}$
(B) $\sin 60^{\circ}$
(C) $\tan 60^{\circ}$
(D) $\sin 30^{\circ}$

Answer

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Precalculus

NCERT Class 10 - Math

Chapter 8

Introduction to Trigonometry

Section 2

$\quad$ Trigonometric Ratios

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Problem 1

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Video Transcript

So in the given question we have a few trigonometry ratios. And we have been given the first ratio which is to tan theta to 10 30 degrees divided by one place times square 30 degree. Right? So and we are asked to find the value from the options signed 60 cause 60 10 16. And we have the fourth option which is signed 30. So one of these options gives us the correct uh answered for this expression. Right? So first what we can do is we can use the identity that one plus 10 square theater is equal to six square theater, right? It is equal to see X squared data. And then we can use the formula that tan Theta is equal to scientific to the weirdly by costa and 60 to is equal to one by cost data. So this will then change to two times scientific to sign 30 liberated by cost 30 divided by one plus stand square feta is six square feet Associate square 30. And from this what we can do is write six square 30 as we can write it as to sign 30 to sign 13. They were read by cost 30 times. We have six square 30 which we can write it as one by cost square 13. So what we would have is they would have two times sine 30 times cost 30. So there is an identity that says sign off to theatre as equal to to sign theater times cost data. So using this identity we can write that two times sine 30 cost 30 is equal to sign off two times 30 which is 16. So the required answer is a sign of 16. So the option that uses the correct answer or the correct value of the given trigonometry ratio is signed 60 option A right now let's move on to the second trigonometry ratio that is given in the question which is one minus and squared 45 degree divided by one place and square 45. Yeah. And here also we have some options. So the option over here is yeah we have option A which has given us 10 of 90 degrees. Then we have option B, which is one option C, which is a sign of 45 degrees and option B S zero. So now we can use the same expression which is one plus tan squared. Theta is equal to six square data in the denominator over here and we can replace trance where 45 degree with science script. 45 degree divided by cost square 40 predicted. So we have one minus sine squared 45 degree divided by cross square 45 degrees divided by one plus stand square 45 years, six square 45 degree, right? According to the identity. So now we can cross multiply and then enumerated to have cost square 45 degree minus sine square 45 degree divided by they crossed where in the there will be a cost square in the denominator times six square 45 can be written last Won by ko square 45 which means the denominator in the denominator cause square 45 divides for square 45 the denominator becomes one. So what we're left with is foursquare 45 minus sine square 45. So now we can use a formula that says cause of two theater can be written us cost square theta minus sine square theater. Mhm. Using which we can write cost square 45 minus sine square 45 is equal to cost of two times 45 which is cost 90. And we know the value of course nineties equal to zero. So the answer for the second econometric ratio is given by option B, which is equal to zero. So in the question we have the third technology ratio and we have a third question which has given us sign of two times A is equal to two times. Sign in is true is true when A is equal to and we have options, option is given us zero degree option being given us 13 degree and option. See she has given us 45 degrees and option D has given us 60 degrees right? So we know that the value of sign of zero is equal to zero, right? So when we substitute A A zero, we will get signed off two times zero in the left hand side which is equal to zero. And on the right hand side we have the left hand side equal to zero. And the right hand side we have two times sine of zero which is also equal to two times zero, which is equal to zero. So the left hand side and the right hand side is equal, which means signed. Two is equal to sign A. Is true when is equal to zero dignities. So this is the required solution of the third question. And now we have 1/4 question which is fourth technologic ratio which has given us two times 10 30 divided by one minus and square data transfer. 30 is equal to want in the options we have cause 16 then we have signed 16 then option C. S 10, 16 and then we have sign of 30. So this is what we have right. So what we're going to do here is we can take the same identities that we used and allowed. Uh huh. We can use an identity that says turn off. Turn of an angle A plus tan of an angle B. Can be calculated using the formula two times. Sorry, 10 of a plus 10 of B. Yeah, divided by one minus than a times Tandy. So how do we use this phone over here? So, we can see that in the issue. That has given the question we have to 10 30. Right? So if that's the case, we can write it us we can write to 10 30 years 10 13 less. 10 30 degrees. Right? And it is divided by one minus dan. 13 times 10 30 which is actually 10 square 30. Which means this is actually can be simplified to the answer and of turn off A plus B. It is not tiny plus 10 beaches 10 of A plus B. All right. So, we shouldn't use the incorrect formula. The formula is 10 of A plus B is equal to 10 of a place to not be. So. Over here, we would have turn off 30 plus 30 as the answer, right? Which is equal to 10 of 60. So now we can look at the options and the option that uses ton of 60 degrees is option C. So option C. Is the required answer. So I hope you understood all the uh answers. Thank you.

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