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Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta?
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Proof:
We need to prove that:
Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta
Step 1: Simplify the expression on the left-hand side
Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta
= Sin theta - Sin cube theta / cos square theta - cos theta / cos cube theta
= Sin theta / cos theta - Sin cube theta / cos cube theta - cos theta / cos cube theta
= (Sin theta / cos theta) - (Sin cube theta + cos cube theta) / (cos cube theta)
Step 2: Use the identity Sin cube theta + cos cube theta = (Sin theta + cos theta)(1 - Sin theta cos theta)
= (Sin theta / cos theta) - (Sin theta + cos theta)(1 - Sin theta cos theta) / (cos cube theta)
= (Sin theta / cos theta) - (Sin theta + cos theta) / (cos square theta) + Sin theta / (cos cube theta)
= (Sin theta / cos theta) + (Sin theta / cos cube theta) - (Sin theta + cos theta) / (cos square theta)
= Sin theta (1 / cos theta + 1 / cos cube theta - 1 / cos square theta - cos theta / cos square theta)
Step 3: Use the identity 1 / cos square theta = 1 + tan square theta
= Sin theta (1 / cos theta + 1 / cos cube theta - (1 + tan square theta) - cos theta / (1 + tan square theta))
= Sin theta (1 / cos theta - cos theta / (1 + tan square theta) + 1 / cos cube theta - tan square theta / (1 + tan square theta))
= Sin theta ((1 - cos square theta) / (cos theta (1 + tan square theta)) + (1 - tan square theta) / (cos cube theta (1 + tan square theta)))
= Sin theta ((Sin square theta / (cos theta (1 + tan square theta))) - (tan square theta Sin square theta / (cos cube theta (1 + tan square theta))))
= Sin theta (Sin theta / (cos theta (1 + tan square theta))) (1 - tan square theta / cos square theta)
= Sin theta / cos theta (1 / (1 + tan square theta))
= tan theta (using the identity Sin theta / cos theta = tan theta)
Therefore, Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta is proved.
We need to prove that:
Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta
Step 1: Simplify the expression on the left-hand side
Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta
= Sin theta - Sin cube theta / cos square theta - cos theta / cos cube theta
= Sin theta / cos theta - Sin cube theta / cos cube theta - cos theta / cos cube theta
= (Sin theta / cos theta) - (Sin cube theta + cos cube theta) / (cos cube theta)
Step 2: Use the identity Sin cube theta + cos cube theta = (Sin theta + cos theta)(1 - Sin theta cos theta)
= (Sin theta / cos theta) - (Sin theta + cos theta)(1 - Sin theta cos theta) / (cos cube theta)
= (Sin theta / cos theta) - (Sin theta + cos theta) / (cos square theta) + Sin theta / (cos cube theta)
= (Sin theta / cos theta) + (Sin theta / cos cube theta) - (Sin theta + cos theta) / (cos square theta)
= Sin theta (1 / cos theta + 1 / cos cube theta - 1 / cos square theta - cos theta / cos square theta)
Step 3: Use the identity 1 / cos square theta = 1 + tan square theta
= Sin theta (1 / cos theta + 1 / cos cube theta - (1 + tan square theta) - cos theta / (1 + tan square theta))
= Sin theta (1 / cos theta - cos theta / (1 + tan square theta) + 1 / cos cube theta - tan square theta / (1 + tan square theta))
= Sin theta ((1 - cos square theta) / (cos theta (1 + tan square theta)) + (1 - tan square theta) / (cos cube theta (1 + tan square theta)))
= Sin theta ((Sin square theta / (cos theta (1 + tan square theta))) - (tan square theta Sin square theta / (cos cube theta (1 + tan square theta))))
= Sin theta (Sin theta / (cos theta (1 + tan square theta))) (1 - tan square theta / cos square theta)
= Sin theta / cos theta (1 / (1 + tan square theta))
= tan theta (using the identity Sin theta / cos theta = tan theta)
Therefore, Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta is proved.
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Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta?
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Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta? 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 Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta?.
Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta? 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 Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that :Sin theta - 2 Sin cube theta / 2 cos cube theta minus cos theta = tan theta?.
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