(1 - A + A)² is equal to

Correct Answer is : 2(1 – sin A)(1 + cos A)

Correct Answer is : 2(1 – sin A)(1 + cos A)

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UPSC NDA I 2015 Mathematics Paper

Question : 38 of 120

Marks: +1, -0

(1 - \sin A + \cos A)^2

is equal to

Solution:

Concept:

Use the identity

\sin^2 A + \cos^2 A = 1

and the square expansion

(a+b)^2 = a^2 + 2ab + b^2

Explanation:

Rewrite

(1 - \sin A + \cos A)^2

[ (1 - \sin A) + \cos A ]^2

Expand:

(1 - \sin A)^2 + \cos^2 A + 2(1 - \sin A)\cos A

Compute

(1 - \sin A)^2 = 1 + \sin^2 A - 2\sin A

Add

\cos^2 A

1 + \sin^2 A - 2\sin A + \cos^2 A = 1 + (\sin^2 A + \cos^2 A) - 2\sin A = 1 + 1 - 2\sin A = 2 - 2\sin A

Now add the cross term:

2(1 - \sin A)\cos A

Total:

2 - 2\sin A + 2(1 - \sin A)\cos A = 2(1 - \sin A) + 2(1 - \sin A)\cos A

Factor

2(1 - \sin A)

2(1 - \sin A)(1 + \cos A)

Answer:

2(1 - \sin A)(1 + \cos A)

, which matches option B.

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