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प्रश्न
Solve the following equation:
`cosec x = 1 + cot x`
उत्तर
Given,
`cosec x = 1 + cot x`
⇒ `1/sin x = 1 + cos x/sin x`
⇒ sin x + cos x = 1
In all such problems we try to reduce the equation in an equation involving single trigonometric expression.
∴ `s 1/sqrt2 sin x + 1/sqrt2 cos x = 1/sqrt2` {dividing by √2 both sides}
⇒ `sin x sin pi/4 + cos pi/4 cos x = cos pi/4.` {cos A cos B + sin A sin B = cos(A − B)}
NOTE: The ratio of sin can also be used in place of cos; the answer stays the same, but the form may change. If you enter numbers for n, you will receive the same values in both forms.
If cos x = cos y, impls x = 2nπ ± y, where n ∈ Z
∴ `x - pi/4 = (2npi ± pi/4).`
∴ `x = (2npi ± pi/4) + pi/4` where n n ∈ Z
`x = 2npi or x = 2npi + pi/2` where n n ∈ Z
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