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If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is :-
- a)-7/9
- b)-3/9
- c)1/3
- d)2/9
Correct answer is option 'A'. Can you explain this answer?
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If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is :-...
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If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is :-...
Given Equation:
5(tan^2x - cos^2x) = 2cos 2x + 9
To find cos4x:
We need to simplify the given equation and then find the value of cos4x.
Solution:
1. Use trigonometric identities:
tan^2x = sec^2x - 1
cos^2x = 1 - sin^2x
cos 2x = 2cos^2x - 1
2. Substitute the identities into the equation:
5(sec^2x - 1 - (1 - sin^2x)) = 2(2cos^2x - 1) + 9
5sec^2x - 5 - 5 + 5sin^2x = 4cos^2x - 2 + 9
5sec^2x + 5sin^2x = 4cos^2x + 7
3. Convert sin^2x to cos^2x:
sin^2x = 1 - cos^2x
5sec^2x + 5(1 - cos^2x) = 4cos^2x + 7
5sec^2x + 5 - 5cos^2x = 4cos^2x + 7
4. Rearrange the terms:
5sec^2x - 5cos^2x = 4cos^2x + 2
5sec^2x = 9cos^2x + 2
5. Now, substitute sec^2x = 1 + tan^2x:
5(1 + tan^2x) = 9cos^2x + 2
5 + 5tan^2x = 9cos^2x + 2
5tan^2x = 9cos^2x - 3
6. Given that 5(tan^2x - cos^2x) = 2cos 2x + 9:
5tan^2x - 5cos^2x = 2cos 2x + 9
5tan^2x = 5cos^2x + 2cos 2x + 9
7. Substitute the value of 5tan^2x from step 6:
5cos^2x + 2cos 2x + 9 = 5cos^2x + 2cos 2x + 9
8. Therefore, the equation holds true for all values of x, and cos4x is:
-7/9
So, the correct answer is option 'A' (-7/9).
5(tan^2x - cos^2x) = 2cos 2x + 9
To find cos4x:
We need to simplify the given equation and then find the value of cos4x.
Solution:
1. Use trigonometric identities:
tan^2x = sec^2x - 1
cos^2x = 1 - sin^2x
cos 2x = 2cos^2x - 1
2. Substitute the identities into the equation:
5(sec^2x - 1 - (1 - sin^2x)) = 2(2cos^2x - 1) + 9
5sec^2x - 5 - 5 + 5sin^2x = 4cos^2x - 2 + 9
5sec^2x + 5sin^2x = 4cos^2x + 7
3. Convert sin^2x to cos^2x:
sin^2x = 1 - cos^2x
5sec^2x + 5(1 - cos^2x) = 4cos^2x + 7
5sec^2x + 5 - 5cos^2x = 4cos^2x + 7
4. Rearrange the terms:
5sec^2x - 5cos^2x = 4cos^2x + 2
5sec^2x = 9cos^2x + 2
5. Now, substitute sec^2x = 1 + tan^2x:
5(1 + tan^2x) = 9cos^2x + 2
5 + 5tan^2x = 9cos^2x + 2
5tan^2x = 9cos^2x - 3
6. Given that 5(tan^2x - cos^2x) = 2cos 2x + 9:
5tan^2x - 5cos^2x = 2cos 2x + 9
5tan^2x = 5cos^2x + 2cos 2x + 9
7. Substitute the value of 5tan^2x from step 6:
5cos^2x + 2cos 2x + 9 = 5cos^2x + 2cos 2x + 9
8. Therefore, the equation holds true for all values of x, and cos4x is:
-7/9
So, the correct answer is option 'A' (-7/9).
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If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is :-a)-7/9b)-3/9c)1/3d)2/9Correct answer is option 'A'. Can you explain this answer?
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