Question

$\sin( \alpha ) = \frac{ {a}^{2} \: - {b}^{2} }{{a}^{2} + {b}^{2}}$
find all the other trigonometric ratios​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\displaystyle{\boxed{\blue{\sf\:\sin\:\theta\:=\:\dfrac{a^2\:-\:b^2}{a^2\:+\:b^2}\:}}}$

$\displaystyle{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}$

$\displaystyle{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}$

$\displaystyle{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}$

$\displaystyle{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}$

$\displaystyle{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}$

Step-by-step-explanation:

Refer to the attachment for the steps.

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Steps to solve the question:

1. Use the identity sin² θ + cos² θ = 1 and write cos² θ as ( 1 - sin² θ ).

2. Use the identity ( a - b )² = a² - 2ab + b² in numerator and ( a + b )² = a² + 2ab + b² in denominator.

3. Simplifying further we get, cos² θ = ( 4a²b² ) / ( a⁴ + 2a²b² + b⁴ ).

4. Factor the above expression and take the square root on both sides, this gives us

$\displaystyle{\underline{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}}$

5. Use the identity tan θ = sin θ / cos θ.

6. Substitute the values of sin θ and cos θ and simplify. This gives us

$\displaystyle{\underline{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}}$

7. Use the identity cosec θ = 1 / sin θ.

8. Substitute the value of sin θ and simplify. This gives us

$\displaystyle{\underline{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}}$

9. Use the identity sec θ = 1 / cos θ.

10. Substitute the value of cos θ and simplify. This gives us

$\displaystyle{\underline{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}}$

11. Use the identity cot θ = 1 / tan θ.

12. Substitute the value of tan θ and simplify. This gives us

$\displaystyle{\underline{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}}$