Question

If tanA=(xsinB) ÷(1-xcosB) and tanB=(ysinA) ÷(1-ycosA), show that sinA/sinB=x/y

No spams please

Answer 1

Step-by-step explanation:

Convert ... Tan A in Cot A and Cot B in tan B then simplify and substitute.

..............

Answer 2

Answer:

sinA/sinB=x/y is proved by using the trigonometric formulae.

Step-by-step explanation:

We are given that

tanA=(xsinB)÷(1-xcosB)

tanB=(ysinA)÷(1-ycosA)

We are asked to show that sinA/sinB=x/y

Simply write tan in terms of cot

$cotA=\frac{1-xcosB}{xsinB}$

Evaluate the fraction. We get,

$cotA=\frac{1}{xsinB}-\frac{xcosB}{xsinB}$

Perform division.

$cotA=\frac{1}{xsinB} -cotB$

Bring cotB to LHS

$cotA+cotB=\frac{1}{xsinB}$  ---------(1)

Similarly convert tanB in terms of cotB.

$cotB=\frac{1-ycosA}{ysinA}$

Elaborate the terms and do division.

$cotB=\frac{1}{ysinA}-\frac{ycosA}{ysinA}$

On performing division for the second term. We get,

$cotB=\frac{1}{ysinA} -cotA$

$cotA+cotB=\frac{1}{ysinA}$   -------(2)

Equate (1) and (2)

$\frac{1}{ysinA} =\frac{1}{xsinB}$

On cross multiplication. We get,

$\frac{x}{y} =\frac{sinA}{sinB}$

Hence proved.