What is (1 + α β)² + (α - β)² equal to?

Correct Answer is : sec2αsec2βsec2α sec2βsec2αsec2β

Correct Answer is : sec2αsec2βsec2α sec2βsec2αsec2β

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UPSC CDS II 2024 Math Paper

Question : 12 of 100

Marks: +1, -0

What is

(1 + \tan\alpha \tan\beta)^{2} + (\tan\alpha - \tan\beta)^{2}

equal to?

Solution:

Concept:

Use algebraic expansion and the trigonometric identity

1 + \tan^2\theta = \sec^2\theta

to simplify the given expression.

Explanation:

Expand

(1 + \tan\alpha \tan\beta)^2

= 1 + 2\tan\alpha \tan\beta + \tan^2\alpha \tan^2\beta

Expand

(\tan\alpha - \tan\beta)^2

= \tan^2\alpha - 2\tan\alpha \tan\beta + \tan^2\beta

Add the two expansions:

(1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2

= 1 + \tan^2\alpha \tan^2\beta + \tan^2\alpha + \tan^2\beta

Group terms:

= 1 + \tan^2\alpha + \tan^2\beta + \tan^2\alpha \tan^2\beta

= (1 + \tan^2\alpha)(1 + \tan^2\beta)

Apply the identity

1 + \tan^2\alpha = \sec^2\alpha

and similarly for

\beta

= \sec^2\alpha \cdot \sec^2\beta

= \sec^2\alpha \sec^2\beta

Answer:

\sec^2\alpha \sec^2\beta

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