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If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A – tan A) (sec B – tan B) (sec C – tan C) then each side is equal to
- a)1
- b)–1
- c)Both A & B
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan...
(secA + tanA)(secB + tanB)(secC + tan C)
=> (secA - tanA)(secB - tanB)(secC - tanC)
{ Mulitply both sides with }
(secA + tanA)(secB + tanB)(secC + tan C)",
(secA + tanA)(secB + tanB)(secC + tan C)",
we get,
(secA + tanA)2(secB + tanB)2(secC + tan C)2
=> (sec2A - tan2A)(sec2B - tan2B)(sec2C - tan2C)
= (1)(1)(1) = 1
=> [(secA + tanA)(secB + tanB)(secC + tanC)]2=1
(secA + tanA)(secB + tanB)(secC + tan C) = ± 1
Similarly, we get
(secA – tanA)(secB – tanB)(secC – tan C) = ± 1
Most Upvoted Answer
If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan...
Tan A) (sec B tan B) (sec C tan C), then we can conclude that sec A tan A = sec B tan B = sec C tan C.
To prove this, let's assume that sec A tan A ≠ sec B tan B. Without loss of generality, let's assume that sec A tan A > sec B tan B.
Then, we have sec A tan A (sec B tan B) (sec C tan C) > sec B tan B (sec B tan B) (sec C tan C).
Simplifying the above expression, we get sec A tan A (sec B tan B) (sec C tan C) > sec B^2 tan^2 B (sec C tan C).
Since sec B^2 tan^2 B = sec B tan B (sec B tan B), we can rewrite the expression as sec A tan A (sec B tan B) (sec C tan C) > sec B tan B (sec B^2 tan^2 B (sec C tan C)).
Using the given equality, we can substitute sec A tan A = sec B tan B, giving us sec B tan B (sec B tan B) (sec C tan C) > sec B tan B (sec B^2 tan^2 B (sec C tan C)).
Cancelling out sec B tan B from both sides, we are left with sec B tan B (sec C tan C) > sec B^2 tan^2 B (sec C tan C).
Dividing both sides by sec C tan C (which is not equal to zero), we get sec B tan B > sec B^2 tan^2 B.
Simplifying the expression, we have 1 > sec B tan B, which is not possible since sec B tan B is always greater than or equal to 1.
Therefore, our assumption that sec A tan A ≠ sec B tan B must be false, and we can conclude that sec A tan A = sec B tan B.
Similarly, we can prove that sec A tan A = sec C tan C.
Hence, if (sec A tan A) (sec B tan B) (sec C tan C) = (sec A tan A) (sec B tan B) (sec C tan C), we can conclude that sec A tan A = sec B tan B = sec C tan C.
To prove this, let's assume that sec A tan A ≠ sec B tan B. Without loss of generality, let's assume that sec A tan A > sec B tan B.
Then, we have sec A tan A (sec B tan B) (sec C tan C) > sec B tan B (sec B tan B) (sec C tan C).
Simplifying the above expression, we get sec A tan A (sec B tan B) (sec C tan C) > sec B^2 tan^2 B (sec C tan C).
Since sec B^2 tan^2 B = sec B tan B (sec B tan B), we can rewrite the expression as sec A tan A (sec B tan B) (sec C tan C) > sec B tan B (sec B^2 tan^2 B (sec C tan C)).
Using the given equality, we can substitute sec A tan A = sec B tan B, giving us sec B tan B (sec B tan B) (sec C tan C) > sec B tan B (sec B^2 tan^2 B (sec C tan C)).
Cancelling out sec B tan B from both sides, we are left with sec B tan B (sec C tan C) > sec B^2 tan^2 B (sec C tan C).
Dividing both sides by sec C tan C (which is not equal to zero), we get sec B tan B > sec B^2 tan^2 B.
Simplifying the expression, we have 1 > sec B tan B, which is not possible since sec B tan B is always greater than or equal to 1.
Therefore, our assumption that sec A tan A ≠ sec B tan B must be false, and we can conclude that sec A tan A = sec B tan B.
Similarly, we can prove that sec A tan A = sec C tan C.
Hence, if (sec A tan A) (sec B tan B) (sec C tan C) = (sec A tan A) (sec B tan B) (sec C tan C), we can conclude that sec A tan A = sec B tan B = sec C tan C.
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If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer?
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If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer?.
If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (sec A + tan A) (sec B + tan B) (sec C + tan C) = (sec A –tan A) (sec B –tan B) (sec C –tan C) then each side is equal toa)1b)–1c)Both A & Bd)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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