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If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2
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If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2?
Proof: 2sinθ - cosθ = 2, given sinθ cosθ = 1/2
Step 1: Rewrite sinθ cosθ = 1/2
We can use the identity sin2θ + cos2θ = 1 to rewrite sinθ cosθ as:
sinθ cosθ = (sinθ cosθ) + (sinθ cosθ) - (sin2θ + cos2θ)
Expanding the right-hand side of this equation, we get:
sinθ cosθ = 2(sinθ cosθ - sin2θ - cos2θ)
Using the given equation sinθ cosθ = 1/2, we can substitute and simplify to get:
1/2 = 2(1/2 - sin2θ - cos2θ)
1/4 = 1 - 2sin2θ - 2cos2θ
Step 2: Rewrite 2sinθ - cosθ
Now we can rewrite 2sinθ - cosθ as:
2sinθ - cosθ = 2(sinθ cosθ + sinθ cosθ) - cosθ
Using the given equation sinθ cosθ = 1/2, we can substitute and simplify to get:
2sinθ - cosθ = (sinθ + cosθ) + (sinθ + cosθ) - cosθ
2sinθ - cosθ = 2sinθ + 2cosθ - cosθ
2sinθ - cosθ = 2sinθ + cosθ
Step 3: Substitute sinθ cosθ = 1/2
Now we can substitute sinθ cosθ = 1/2 into our rewritten expression for 2sinθ - cosθ:
2sinθ - cosθ = 2(sinθ cosθ) + cosθ
2sinθ - cosθ = 1 + cosθ
Step 4: Use given equation sin2θ + cos2θ = 1
Finally, we can use the given equation sinθ 2 cosθ = 1 to rewrite cosθ as:
cosθ = 1/sinθ 2
Substituting this expression for cosθ into our previous equation, we get:
2sinθ - cosθ = 1 + 1/sinθ 2
2sinθ - 1/sinθ 2 = 2
Multiplying both sides by sinθ 2, we get:
2(sin2θ - 1) = 2sinθ 2
2sin2θ - 4 = 0
sin2θ = 2
Step 5: Conclusion
We have shown that if sinθ cosθ = 1/2, then 2sinθ - cosθ
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If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2?
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If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2?.
If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sintheta+2 costheta =1 prove that 2 sin theta-cos theta =2?.
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