Express cos A and tan A in terms of sin A.

Q) Express cos A and tan A in terms of sin A.

(Q 25 B – 30/2/3 – CBSE 2026 Question Paper)

Ans:

Step 1: We know that according to trigonometric identity:

sin ² θ + cos ² θ = 1

or for θ = A, we can say, sin ² A + cos ² A = 1

∴ cos ² A = 1 – sin ² A

∴ cos A = $\pm\sqrt{1-\sin^2 A}$

Here, sign value of cos A will be based on the quadrant in which angle A lies.

Therefore, in terms of sin A, cos A can be expressed as $\pm\sqrt{1-\sin^2 A}$ .

Step 2: We know that tan A is expressed by: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

or for θ = A, we can say, $\tan A = \frac{\sin A}{\cos A}$

By substituting value of cos A from step 1, we get:

∵ $\tan A = \frac{\sin A}{\cos A}$

∴ $\tan A = \frac{\sin A}{\pm\sqrt{1-\sin^2 A}}$

∴ $\tan A = \frac{\pm \sin A}{\sqrt{1-\sin^2 A}}$

Here, sign value of tan A will be based on the quadrant in which angle A lies

Therefore, in terms of sin A, cos A can be expressed as $\frac{\pm \sin A}{\sqrt{1-\sin^2 A}}$ .

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