Trigonometric Identities

Geometry Level 2

If cosecA - cotA = 1/3 , find the value of (cosecA + cotA) .

Please add solution if you know the solution.


The answer is 3.

Sudo Jarvis by Sudo Jarvis , 21, India

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2 solutions

( csc ⁡ A − cot ⁡ ( A ) ) ∗ ( csc ⁡ ( A ) + cot ⁡ ( A ) ) = csc ⁡ 2 ( A ) − cot ⁡ 2 ( A ) = 1 (\csc{A} - \cot(A)) * (\csc(A) + \cot(A)) = \csc^{2}(A) - \cot^{2}(A) = 1 .

So if csc ⁡ ( A ) − cot ⁡ ( A ) = 1 3 \csc(A) - \cot(A) = \dfrac{1}{3} we must have csc ⁡ ( A ) + cot ⁡ ( A ) = 3 \csc(A) + \cot(A) = \boxed{3} .

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( csc ⁡ ( A ) − cot ⁡ ( A ) ) ( csc ⁡ ( A ) − cot ⁡ ( A ) ) = 1 3 ( csc ⁡ ( A ) + cot ⁡ ( A ) (\csc (A) - \cot (A))(\csc (A) - \cot (A)) = \frac{1}{3}(\csc(A) + \cot (A) . Since csc ⁡ 2 ( A ) − cot ⁡ 2 ( A ) = 1 \csc^{2}(A) - \cot^{2}(A) = 1 , then we must have csc ⁡ ( A ) + cot ⁡ ( A ) = 3 \csc(A) + \cot (A) = \boxed{3} .

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