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Simplify expression sec×sin(-×)+tan(-x) over 1+sec(-×)

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5 votes

Answer:

-tan(x)/sin^2(x).

Explanation:

To simplify the expression:

sec(x)sin(-x) + tan(-x)

1 + sec(-x)

We can use the fact that sin(-x) = -sin(x), cos(-x) = cos(x), and tan(-x) = -tan(x) to rewrite the expression as:

-sec(x)sin(x) - tan(x)

1 + sec(x)

Next, we can multiply both the numerator and denominator by the conjugate of the denominator, 1 - sec(x), to eliminate the square root in the denominator. This gives:

(-sec(x)sin(x) - tan(x))(1 - sec(x))

(1 + sec(x))(1 - sec(x))

Simplifying the numerator by distributing and combining like terms, we get:

-sec(x)sin(x) + sec(x)tan(x) + tan(x) - sec(x)tan(x)

1 - sec^2(x)

Simplifying further by canceling out the sec(x)tan(x) terms in the numerator, we get:

-tan(x)

1 - sec^2(x)

Finally, we can use the identity 1 - sec^2(x) = sin^2(x) to rewrite the denominator and simplify further:

-tan(x)

sin^2(x)

Therefore, the simplified expression is -tan(x)/sin^2(x).

User Manishi
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