Simplify: (1+sec(-x))/(sin(-x)+tan(-x))=-csc x

Simplify using trigonometric identities: Simplify the left-hand side using trigonometric identities; sec(-x) = sec x, sin(-x) = -sin x, tan(-x) = -tan x. Calculation: (1 + sec x)/(-sin x - tan x)Factor out negative sign: Rewrite the expression by factoring out a negative sign from the denominator. Calculation: (1 + sec x)/(-(sin x + tan x)) = -(1 + sec x)/(sin x + tan x)Rewrite using identities: Use the identity sec x = 1/cos x and tan x = sin x/cos x to rewrite sec x and tan x. Calculation: -(1 + 1/cos x)/(sin x + sin x/cos x)Simplify denominator: Simplify the denominator by getting a common denominator. Calculation: -(1 + 1/cos x)/(sin x * (1 + 1/cos x)/cos x)Cancel out common terms: Cancel out (1 + 1/cos x) in the numerator and denominator. Calculation: -cos x/sin x

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Simplify: (1+sec(-x))/(sin(-x)+tan(-x))=-csc x

Simplify: $\frac{1+\sec(-x)}{\sin(-x)+\tan(-x)}=-\csc x$

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Algebra 2

Sum of finite series starts from 1

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Q. Simplify:

\frac{1+\sec(-x)}{\sin(-x)+\tan(-x)}=-\csc x

Simplify using trigonometric identities: Simplify the left-hand side using trigonometric identities; $\sec(-x) = \sec x$ , $\sin(-x) = -\sin x$ , $\tan(-x) = -\tan x$ . Calculation: $\frac{1 + \sec x}{-\sin x - \tan x}$
Factor out negative sign: Rewrite the expression by factoring out a negative sign from the denominator. $\newline$ Calculation: $(1 + \sec x)/(-(\sin x + \tan x)) = -(1 + \sec x)/(\sin x + \tan x)$
Rewrite using identities: Use the identity $\sec x = \frac{1}{\cos x}$ and $\tan x = \frac{\sin x}{\cos x}$ to rewrite $\sec x$ and $\tan x$ . $\newline$ Calculation: $-\left(\frac{1 + \frac{1}{\cos x}}{\sin x + \frac{\sin x}{\cos x}}\right)$
Simplify denominator: Simplify the denominator by getting a common denominator. $\newline$ Calculation: $-\frac{1 + \frac{1}{\cos x}}{\sin x \cdot \left(1 + \frac{1}{\cos x}\right)/\cos x}$
Cancel out common terms: Cancel out $(1 + \frac{1}{\cos x})$ in the numerator and denominator. $\newline$ Calculation: $-\frac{\cos x}{\sin x}$