Step-by-step explanation:
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Answer:
[tex]\displaystyle{\boxed{\blue{\sf\:\sin\:\theta\:=\:\dfrac{a^2\:-\:b^2}{a^2\:+\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}[/tex]
[tex]\displaystyle{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}[/tex]
[tex]\displaystyle{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}[/tex]
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
[tex]\displaystyle{\underline{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}}[/tex]
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
[tex]\displaystyle{\underline{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}}[/tex]
7. Use the identity cosec θ = 1 / sin θ.
8. Substitute the value of sin θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}}[/tex]
9. Use the identity sec θ = 1 / cos θ.
10. Substitute the value of cos θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}}[/tex]
11. Use the identity cot θ = 1 / tan θ.
12. Substitute the value of tan θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}}[/tex]
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Verified answer
Step-by-step explanation:
please mark me brainlist
Answer:
[tex]\displaystyle{\boxed{\blue{\sf\:\sin\:\theta\:=\:\dfrac{a^2\:-\:b^2}{a^2\:+\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}[/tex]
[tex]\displaystyle{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}[/tex]
[tex]\displaystyle{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}[/tex]
[tex]\displaystyle{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}[/tex]
Step-by-step-explanation:
Refer to the attachment for the steps.
─────────────────────
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
[tex]\displaystyle{\underline{\boxed{\pink{\sf\:\cos\:\theta\:=\:\dfrac{2ab}{a^2\:+\:b^2}\:}}}}[/tex]
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
[tex]\displaystyle{\underline{\boxed{\green{\sf\:\tan\:\theta\:=\:\dfrac{a^2\:-\:b^2}{2ab}\:}}}}[/tex]
7. Use the identity cosec θ = 1 / sin θ.
8. Substitute the value of sin θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\red{\sf\:cosec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{a^2\:-\:b^2}\:}}}}[/tex]
9. Use the identity sec θ = 1 / cos θ.
10. Substitute the value of cos θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\orange{\sf\:\sec\:\theta\:=\:\dfrac{a^2\:+\:b^2}{2ab}\:}}}}[/tex]
11. Use the identity cot θ = 1 / tan θ.
12. Substitute the value of tan θ and simplify. This gives us
[tex]\displaystyle{\underline{\boxed{\purple{\sf\:\cot\:\theta\:=\:\dfrac{2ab}{a^2\:-\:b^2}\:}}}}[/tex]