If cotθ+tanθ=x and secθ-cosθ=y, then (a) sinθcosθ=(1)/(x) (b) sinθtanθ=y (d) (x^2 y)^13+(x y^2)^

step 1 So in this problem we need to find the correct option. So I'm solving one by one. So chord theta

If cotθ+tanθ=x and secθ-cosθ=y, then (a) sinθcosθ=(1)/(x)...

If $\cot \theta+\tan \theta=x$ and $\sec \theta-\cos \theta=y$, then
(a) $\sin \theta \cos \theta=\frac{1}{x}$
(b) $\sin \theta \tan \theta=y$
(d) $\left(x^{2} y\right)^{13}+\left(x y^{2}\right)^{1 / 3}=1 .$
(c) $\left(x^{2} y\right)^{23}-\left(x y^{2}\right)^{23}=1$

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Key Concepts

Algebraic Manipulation of Trigonometric Expressions

Once trigonometric functions are expressed in a uniform way and identities are applied, algebraic techniques such as factoring, exponentiation, and taking reciprocals become essential in deducing further relationships among the expressions.

Trigonometric Representations

This concept involves rewriting functions like cotangent, tangent, secant, and cosine in terms of sine and cosine. Converting all expressions to a common basis simplifies the analysis and manipulation of trigonometric equations.

Reciprocal and Quotient Identities

These identities express functions such as cot?, tan?, and sec? in terms of sine and cosine (e.g., cot? = cos?/sin? and tan? = sin?/cos?). They are crucial for transforming complex trigonometric expressions into simpler forms that can be managed with algebraic operations.

Pythagorean Identity

The identity sin²? + cos²? = 1 is a fundamental relation in trigonometry. It is often used to substitute and simplify expressions after rewriting trigonometric functions in terms of sine and cosine.

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