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jmc

algebra intermediate

Problem

The numbers

sin x,

cos x,

tan x

form a geometric sequence. Find

cos^{3} x + cos^{2} x .

Solution

We have that

cos^{2} x = sin x tan x = sin x \cdot \frac{sin x}{cos x} = \frac{sin ^{2} x}{cos x} .

Then

cos^{3} x = sin^{2} x = 1 - cos^{2} x,

so

cos^{3} x + cos^{2} x = 1 .

Final answer

1