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If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}?
Most Upvoted Answer
If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ ...
Sec²Q – tan²Q = 1
(3x)² – (3/x)² = 1
9x² – 9/x² = 1
9×(x² – 1/x²) = 1
That's all🙂
(3x)² – (3/x)² = 1
9x² – 9/x² = 1
9×(x² – 1/x²) = 1
That's all🙂
Community Answer
If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ ...
Solution:
Given, 3x = sec θ and 3/x = tan θ
Step 1: Finding the values of sin θ and cos θ
We know that, sec θ = 1/cos θ
So, 3x = 1/cos θ
Or, cos θ = 1/3x
And, tan θ = sin θ/cos θ
So, 3/x = sin θ/(1/3x)
Or, sin θ = 9/x²
Now, we have values of sin θ and cos θ.
Step 2: Finding the value of 9{x² - 1/ x²}
We know that, sin² θ + cos² θ = 1
Substituting the values of sin θ and cos θ, we get
(9/x²)² + (1/3x)² = 1
Simplifying the equation, we get
81/x⁴ + 1/9x² = 1
Multiplying both sides by 9x⁴, we get
81 + x⁴ = 9x⁶
Or, 9x⁶ - x⁴ - 81 = 0
Now, we can find the value of 9{x² - 1/ x²} using the formula (a³ - b³) = (a - b)(a² + ab + b²)
Let a = 3x² and b = 1/3x², then
9(x⁴ - 1/9) = 9[(3x²)² - (1/3x²)²]
= 9[(3x² - 1/3x²)(9x⁴ + 1 + 1/9x⁴)]
= 9[(9x⁴ - 1)(9x⁴ + 1 + 1/9x⁴)]
= 9(81x⁸ + 9x⁴ - 1)
Step 3: Conclusion
Therefore, the value of 9{x² - 1/ x²} is 81x⁸ + 9x⁴ - 1.
Given, 3x = sec θ and 3/x = tan θ
Step 1: Finding the values of sin θ and cos θ
We know that, sec θ = 1/cos θ
So, 3x = 1/cos θ
Or, cos θ = 1/3x
And, tan θ = sin θ/cos θ
So, 3/x = sin θ/(1/3x)
Or, sin θ = 9/x²
Now, we have values of sin θ and cos θ.
Step 2: Finding the value of 9{x² - 1/ x²}
We know that, sin² θ + cos² θ = 1
Substituting the values of sin θ and cos θ, we get
(9/x²)² + (1/3x)² = 1
Simplifying the equation, we get
81/x⁴ + 1/9x² = 1
Multiplying both sides by 9x⁴, we get
81 + x⁴ = 9x⁶
Or, 9x⁶ - x⁴ - 81 = 0
Now, we can find the value of 9{x² - 1/ x²} using the formula (a³ - b³) = (a - b)(a² + ab + b²)
Let a = 3x² and b = 1/3x², then
9(x⁴ - 1/9) = 9[(3x²)² - (1/3x²)²]
= 9[(3x² - 1/3x²)(9x⁴ + 1 + 1/9x⁴)]
= 9[(9x⁴ - 1)(9x⁴ + 1 + 1/9x⁴)]
= 9(81x⁸ + 9x⁴ - 1)
Step 3: Conclusion
Therefore, the value of 9{x² - 1/ x²} is 81x⁸ + 9x⁴ - 1.
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If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}?
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If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}?.
If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 3x=sec theta and 3/x= tan theta, then find the value of 9{x^2 - 1/ x^2}?.
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