Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.cot(-θ)/sec(-θ)

Accepted Answer

Recall the definitions and properties of the trigonometric functions involved: $$ \cot 	heta = \frac{\cos 	heta}{\sin 	heta} $$ and $$ \sec 	heta = \frac{1}{\cos 	heta} . $$Also, remember the even-odd properties: $$ \cos(-	heta) = \cos 	heta  ($$even function) and $$ \sin(-	heta) = -\sin 	heta  ($$odd function). Rewrite $$ \cot(-	heta) $$ using the definition of cotangent and the odd-even properties: $$ \cot(-	heta) = \frac{\cos(-	heta)}{\sin(-	heta)} = \frac{\cos 	heta}{-\sin 	heta} = -\frac{\cos 	heta}{\sin 	heta} . $$Rewrite $$ \sec(-	heta) $$ using the definition of secant and the even property of cosine: $$ \sec(-	heta) = \frac{1}{\cos(-	heta)} = \frac{1}{\cos 	heta} . $$Form the quotient $$ \frac{\cot(-	heta)}{\sec(-	heta)}  by $$substituting the expressions from steps 2 and 3: $$ \frac{-\frac{\cos 	heta}{\sin 	heta}}{\frac{1}{\cos 	heta}} . $$Simplify the complex fraction by multiplying numerator and denominator appropriately to eliminate the quotient: $$ -\frac{\cos 	heta}{\sin 	heta} 	imes \frac{\cos 	heta}{1} = -\frac{\cos^2 	heta}{\sin 	heta} . $$This expression is now in terms of $$ \sin 	heta $$ and $$ \cos 	heta $$ only, with no quotients involving other trigonometric functions.

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Even and Odd Trigonometric Functions

Trigonometric Identities for Cotangent and Secant

Simplification of Trigonometric Expressions