How to verify Trigonometric Identities?
To proof and verify the identities we will make use of the basic trigonometric identities to make sure that both the sides of the equation is equal to each other.
1. If tan A = (sin θ
- cos θ)/(sin θ + cos θ) then prove that,
sin θ + cos θ = ± √2 cos A
Solution:
We know that, sec^{2} A = 1 + tan^{2} ANow taking square root on both the sides we get,
sin θ + cos θ = ± √2 cos A .
Proved
More examples to get the basic ideas to proof and verify Trigonometric Identities.
Proved
\(\frac{cos α}{x} = \frac{sin α}{2y} = \frac{\sqrt{cos^{2} α + sin^{2} α}}{x^{2} + 4y^{2}} = \frac{1}{x^{2} + 4y^{2}}
\)
\(Therefore, cos θ = \frac{x}{x^{2} + 4y^{2}} and sin θ = \frac{2y}{x^{2} + 4y^{2}}\)
Now, 2x sec α - y csc α = 3
⇒ 2x ∙ \(\frac{1}{cos α}\) - y ∙ \(\frac{1}{sin α}\) = 3, [Since, sec α = \(\frac{1}{cos α}\) and csc α = \(\frac{1}{sin α}] \)
⇒ 2x ∙ \(\frac{\sqrt{x^{2} + 4y^{2}}}{x}\) - y ∙ \(\frac{\sqrt{x^{2} + 4y^{2}}}{2y}\) = 3, [putting the values of sin α and cos α]
⇒ \(\frac{3}{2}\sqrt{x^{2} + 4y^{2}} = 3\)
⇒ \(\sqrt{x^{2} + 4y^{2}} = 2\)
Now taking square root on both the sides
we get,
Proved
Note: Remember there is no set method that can be applied to verify trigonometric identities. However, a few different techniques needed to follow to start verifying from one side, based on the identity which is to be verified.
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