Simplify cos(-t)+cos(-t)tan(-t)^2

Problem

cos(−t)+cos(−t)*tan^2(−t)

Solution

1. Apply even and odd identities to simplify the trigonometric functions with negative arguments.

cos(−t)=cos(t)

tan(−t)=−tan(t)

tan^2(−t)=(−tan(t))^2=tan^2(t)

2. Rewrite the expression using the simplified terms.

cos(t)+cos(t)*tan^2(t)

3. Factor out the common term cos(t) from the expression.

cos(t)*(1+tan^2(t))

4. Apply the Pythagorean identity 1+tan^2(t)=sec^2(t) to simplify the term inside the parentheses.

cos(t)*sec^2(t)

5. Use the reciprocal identity sec(t)=1/cos(t) to rewrite the expression in terms of cosine.

cos(t)⋅1/cos^2(t)

6. Simplify the fraction by canceling one factor of cos(t)

1/cos(t)

7. Apply the reciprocal identity again to reach the simplest form.

sec(t)

Final Answer

cos(−t)+cos(−t)*tan^2(−t)=sec(t)

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