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Question

If tanθ=512, find (1+cos θ)×(1sinθ)(1+sinθ)×(1cosθ).

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Solution


Consider ΔABC, right angled at B.

Given: tanθ=512

tanθ=ABBC=512

If AB = 5k, then BC = 12k, where k is a positive number.

Applying Pythagoras theorem, we have

AB2+BC2=AC2

AC2=(12k)2+(5k)2

AC2=169k2

AC=13k

sin θ=ABAC=5k13k=513

cos θ=BCAC=12k13k=1213

(1+cos θ)×(1sin θ)(1+sin θ)×(1cos θ)=(1+1213)×(1513)(1+513)×(11213)

=2513×8131813×113

=20018=1009

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