"Simplify the trigonometric expression sin x+cot x cos x"

Recognize Identities: Recognize the trigonometric identities involved. The expression contains sin x, cot x, and cos x. We know that cot x is the reciprocal of tan x, which means cot x = 1/tan x or cot x = cos x/sin x.Substitute Expression: Substitute the expression for cot x in terms of sin x and cos x. sin x + cot x cos x = sin x + (cos x/sin x) cos xSimplify by Multiplying: Simplify the expression by multiplying cos x with cos x/sin x. sin x + (cos x/sin x) cos x = sin x + (cos^2 x/sin x)Combine Terms: Combine the terms over a common denominator. sin x + (cos^2 x/sin x) = (sin^2 x + cos^2 x)/sin xRecognize Another Identity: Recognize another trigonometric identity. We know that sin^2 x + cos^2 x = 1, which is the Pythagorean identity for sine and cosine.Apply Pythagorean Identity: Apply the Pythagorean identity to the expression. (sin^2 x + cos^2 x)/sin x = 1/sin xRecognize Reciprocal: Recognize that 1/sin x is the reciprocal of sin x, which is csc x. 1/sin x = csc xWrite Final Expression: Write the final simplified expression. The simplified expression is csc x.

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Simplify the trigonometric expression $\newline$ $\sin x + \cot x \cos x$

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Grade 7

Add and subtract like terms: with exponents

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Q. Simplify the trigonometric expression

\newline

\sin x + \cot x \cos x

Recognize Identities: Recognize the trigonometric identities involved. $\newline$ The expression contains $\sin x$ , $\cot x$ , and $\cos x$ . We know that $\cot x$ is the reciprocal of $\tan x$ , which means $\cot x = \frac{1}{\tan x}$ or $\cot x = \frac{\cos x}{\sin x}$ .
Substitute Expression: Substitute the expression for $\cot x$ in terms of $\sin x$ and $\cos x$ . $\sin x + \cot x \cos x = \sin x + \left(\frac{\cos x}{\sin x}\right) \cos x$
Simplify by Multiplying: Simplify the expression by multiplying $\cos x$ with $\frac{\cos x}{\sin x}$ . $\sin x + \left(\frac{\cos x}{\sin x}\right) \cos x = \sin x + \left(\frac{\cos^2 x}{\sin x}\right)$
Combine Terms: Combine the terms over a common denominator. $\newline$ $\sin x + \frac{\cos^2 x}{\sin x}$ $\newline$ = $\frac{\sin^2 x + \cos^2 x}{\sin x}$
Recognize Another Identity: Recognize another trigonometric identity. $\newline$ We know that $\sin^2 x + \cos^2 x = 1$ , which is the Pythagorean identity for sine and cosine.
Apply Pythagorean Identity: Apply the Pythagorean identity to the expression. $\newline$ $(\sin^2 x + \cos^2 x)/\sin x = 1/\sin x$
Recognize Reciprocal: Recognize that $\frac{1}{\sin x}$ is the reciprocal of $\sin x$ , which is $\csc x$ . $\newline$ $\frac{1}{\sin x} = \csc x$
Write Final Expression: Write the final simplified expression. $\newline$ The simplified expression is $\csc x$ .