solve the trigonometric identity
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Hi Maths Lover!
Given: A Trigonometric Expression [tex]\frac{cos(A)}{1-sin(A)}-tan(A)[/tex]
To Find: The solution of the trigonometric Expression [tex]\frac{cos(A)}{1-sin(A)}-tan(A)[/tex]
Basic Concept[s]:
•} Trigonometry - The branch called “Trigonometry” basically deals with the study of the relationship between the sides and angles of the right-angle triangle.
•} Trigonometric Ratios - Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
•} cos - Cosine (or cos) is the trigonometric function that is equal to the ratio of the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse.
•} sin - Sine (or sine) is the trigonometric function that is equal to the ratio of the side opposite a given angle (in a right-angled triangle) to the hypotenuse.
•} tan - Tangent (or tan) is the ratio of the side opposite the angle we know, or want to know, over the side adjacent to that angle. We can write [tex]tan \theta = \frac{sin\theta}{cos \theta}[/tex].
•} sec - Secant (or sec) is the length of the hypotenuse divided by the length of the adjacent side. We can write [tex]sec \theta = \frac{1}{cos \theta}[/tex].
•} A Trigonometric identity states that [tex]sin^{2} \theta + cos^{2} \theta = 1[/tex].
Solution:
[tex]\frac{cos(A)}{1-sin(A)}-\frac{sin (A)}{cos (A)}[/tex]
[tex]\frac{cos(A)^{2}-(1-sin(A)) \times sin(A)}{(1-sin(A)) \times cos(A)}[/tex]
[tex]\frac{cos(A)^{2}-(sin(A)-sin(A)^{2})}{(1-sin(A)) \times cos(A)}[/tex]
[tex]\frac{cos(A)^{2}-sin(A)+sin(A)^{2}}{(1-sin(A)) \times cos(A)}[/tex]
[tex]\frac{cos(A)^{2}+sin(A)^{2}-sin(A)}{(1-sin(A)) \times cos(A)}[/tex]
[tex]\frac{1-sin(A)}{(1-sin(A)) \times cos(A)}[/tex]
[tex]\frac{1}{cos(A)}[/tex]
= [tex]sec \theta[/tex]
Therefore, the solution of the trigonometric expression [tex]\frac{cos(A)}{1-sin(A)}-tan(A)[/tex] is [tex]sec \theta[/tex].
Final Answer:
Hence, the solution of the trigonometric expression [tex]\frac{cos(A)}{1-sin(A)}-tan(A)[/tex] is [tex]sec \theta[/tex].
Hope it helps!
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