Proving the Trigonometric Identity: SecA tanA / cscA cotA (cscA - cotA) / (secA - tanA)
Trigonometric identities play a crucial role in simplifying complex expressions and solving equations in mathematics. One such interesting identity is:
SecA tanA / cscA cotA (cscA - cotA) / (secA - tanA)
Proof:
Rewrite the Left Side:Start with the expression:
SecA tanA / cscA cotA
Using the definitions of the trigonometric functions: secA 1 / cosA tanA sinA / cosA cscA 1 / sinA cotA cosA / sinA Substitute the trigonometric identities:secA tanA (1 / cosA) * (sinA / cosA) (1 sinA) / cosA
cscA cotA (1 / sinA) * (cosA / sinA) (1 cosA) / sinA
Thus, the left side becomes:
(1 sinA / sinA) / (1 cosA / cosA) (1 sinA sinA) / (1 cosA cosA)
Rewrite the Right Side:Now let's rewrite the right side:
(cscA - cotA) / (secA - tanA)
Using the definitions: cscA - cotA (1 / sinA) - (cosA / sinA) (1 cosA) / sinA secA - tanA (1 / cosA) - (sinA / cosA) (1 sinA) / cosA Substitute these into the right side:((1 cosA) / sinA) / ((1 sinA) / cosA) (1 cosA cosA) / (1 sinA sinA)
Set Both Sides Equal:Now we have the simplified forms:
Left Side:
(1 sinA sinA) / (1 cosA cosA)
Right Side:
(1 cosA cosA) / (1 sinA sinA)
Cross-Multiply:To prove the equality, we can cross-multiply:
(1 sinA) (1 sinA) (1 cosA) (1 cosA)
Which simplifies to:
1 sin2A 1 cos2A
Using the Pythagorean identity:
1 - sin2A cos2A
And 1 - cos2A sin2A
Both sides are equal:
cos2A sin2A sin2A cos2A
Therefore, the identity is proven.
Conclusion:We have shown that:
SecA tanA / cscA cotA (cscA - cotA) / (secA - tanA)
Summary:
The problem of verifying the identity SecA tanA / cscA cotA equating to (cscA - cotA) / (secA - tanA) involves expressing trigonometric functions in terms of sine and cosine and using Pythagorean identities to simplify and prove the equality.