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How do you simplify #(sin^2(t)+cos^2(t) )/ cos^2(t)#?

Answer 1

#sec^2t#

Divide each term on the numerator by #cos^2t#

#rArr(sin^2t)/cos^2t+(cos^2t)/cos^2t=tan^2t+1........ (A)#

From the identity #color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2t+cos^2t=1)color(white)(a/a)|)))# divide each term by #cos^2t#

#(sin^2t)/cos^2t+(cos^2t)/cos^2t=1/cos^2t# we obtain the identity.

#color(red)(|bar(ul(color(white)(a/a)color(black)(tan^2t+1=sec^2t)color(white)(a/a)|)))# Substitute this result into the right side of (A)

#rArr(sin^2t+cos^2t)/cos^2t=sec^2t#

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Answer 2

Charles Autterson

To simplify (sin²(t) + cos²(t)) / cos²(t), you can use the Pythagorean identity, which states that sin²(t) + cos²(t) = 1. Therefore, the expression simplifies to 1 / cos²(t). Since 1 / cos(t) = sec(t), the final simplified expression is sec²(t).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.