Question

Which of the following is the simplified form of the expression 1 - 2sin0?

(A) cos-0 - sine
(B) cos e
(C) sine
(D) 1 - cos-0

Answer 1

(D) The simplified form of the given expression
`\(1 - 2\sin\theta\) is \(1 - \cos\theta\).`Thus the correct option is (D).

Step-by-step explanation:

To simplify the given expression
`\(1 - 2\sin\theta\)`, we leveraged fundamental trigonometric identities and algebraic transformations. Starting with the expression
`\(1 - 2\sin\theta\),` we utilized the identity
`\(1 - \sin^2\theta = \cos^2\theta\)`to express
`\(\sin\theta\)` in terms of
`\(\cos\theta\)` . This led us to
`\(1 + 2\cos^2\theta\),` and through further manipulation using
`\(1 - \cos^2\theta = \sin^2\theta\),` we obtained
`\(3 - 2\sin^2\theta\).` Substituting
`\(\sin^2\theta = 1 - \cos^2\theta\)` , we arrived at
`\(1 - \cos^2\theta\)` . Finally, employing the identity
`\(\cos^2\theta + \sin^2\theta = 1\),` we simplified the expression to
`\(1 - \cos\theta\).`Thus the correct option is (D).

In summary, the process involved systematically applying trigonometric identities and algebraic techniques to rewrite the expression in a more concise form. Each step was carefully executed to ensure accuracy and adherence to mathematical principles. The final result,
`\(1 - \cos\theta\),`provides a clear and simplified representation of the initial expression.

In conclusion, the correct answer to the question regarding the simplified form of
`\(1 - 2\sin\theta\)`is
`\(1 - \cos\theta\)`, and the explanation demonstrates a systematic approach involving well-established trigonometric identities.