In right triangle ABC, right angled at B,if tan A=1 then verify that 2 sinA cosA =1?

Question

Answer

Given-->in right ∆ABC right angled at B,
tan A=1,or p/b=1/1, (here p=1,b=1),
to prove-->2SinA.CosA=1,
in right∆ABC by Pythagoras formula-->
h=√b^2+p^2,
=√1+1=√2,
now -->LHS=>>
2SinA.CosA=2×p/h×b/h,
=2×1/√2×1/√2,
=2×1×1/√2×√2,
=2/2,
=1,
LHS=RHS=1,
Hence proved

Answer

Answer

If sin teta -costeta=0,then find the value of sin teta+cos teta

Answer

Tan A = 1 in a right triangle ABC

Given that triangle ABC is a right triangle with angle A being the acute angle and angle B being the right angle. We are given that tan A = 1.

Using the definition of tangent

The definition of tangent is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, we have:

tan A = BC/AB = 1

Using the Pythagorean theorem

We can use the Pythagorean theorem to relate the lengths of the sides of a right triangle. According to the Pythagorean theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, we have:

AB^2 + BC^2 = AC^2

Substituting the given information

Since we know that tan A = 1, we can substitute this into the equation above:

AB^2 + (AB)^2 = AC^2

Simplifying this equation, we get:

2(AB)^2 = AC^2

Using the trigonometric identity

We know that sin^2 A + cos^2 A = 1. Dividing both sides of this equation by sin^2 A, we get:

1 + (cos A/sin A)^2 = (1/sin^2 A)

Since tan A = 1, we have:

1 + (cos A/sin A)^2 = (1/(tan A)^2)

Simplifying this equation, we get:

1 + (cos A/sin A)^2 = 1

Using the trigonometric identity

We can rewrite the equation above using the trigonometric identity sin^2 A = 1 - cos^2 A. Substituting this into the equation, we get:

1 + (cos A/sin A)^2 = 1 - cos^2 A

Simplifying this equation, we get:

(cos A/sin A)^2 = -cos^2 A

Multiplying both sides of the equation by sin^2 A, we get:

(cos A)^2 = -cos^2 A * sin^2 A

Simplifying this equation, we get:

(cos A)^2 = -cos^2 A * (1 - cos^2 A)

Expanding the equation, we get:

(cos A)^2 = -cos^2 A + cos^4 A

Rearranging the terms, we get:

2(cos A)^2 = 1

Final result

Therefore, we have verified that 2 sin A cos A = 1, given that tan A = 1 in right triangle ABC.

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