In a right triangle ABC, right-angled at B, if $\tan A = 1$, then verify that $2 \sin A \cos A = 1$.

Step 1: Define the triangle sides using tan A

Given that $\tan A = 1$ in a right-angled triangle ABC, where the right angle is at B. We know that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, $BC$ (opposite to A) divided by $AB$ (adjacent to A) equals 1.

Step 2: Express sides in terms of a constant

Since $\frac{BC}{AB} = 1$, it implies that $BC = AB$. We can let both sides be equal to a constant $k$, where $k$ is a positive number, for easier calculation.

Step 3: Calculate the hypotenuse using Pythagoras theorem

Now we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse ($AC$) is equal to the sum of the squares of the other two sides ($AB$ and $BC$). Substituting $AB=k$ and $BC=k$, we find the length of the hypotenuse $AC$.

Step 4: Calculate sin A

The sine of angle A is the ratio of the length of the side opposite to angle A ($BC$) to the length of the hypotenuse ($AC$). Substituting the values we found, we calculate $\sin A$.

Step 5: Calculate cos A

The cosine of angle A is the ratio of the length of the side adjacent to angle A ($AB$) to the length of the hypotenuse ($AC$). Substituting the values we found, we calculate $\cos A$.

Step 6: Verify the expression $2 \sin A \cos A = 1$

Finally, we substitute the calculated values of $\sin A$ and $\cos A$ into the given expression $2 \sin A \cos A$. We perform the multiplication to show that the expression evaluates to 1, thus verifying the statement. This matches the method shown in NCERT textbook example 4 where $\tan A=1$ is used to verify $2 \sin A \cos A = 1$.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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