Question

If sin theta + cos theta=√3, then prove that tan theta +cot theta=1

Answer 1

Question:

If sin@ + cos@ = √3 , then prove that

tan@ + cot@ = 1.

Note:

✓ tanA = sinA/cosA

✓ cotA = cosA/sinA

✓ (sinA)^2 + (cosA)^2 = 1

✓ sin(A+B) = sinA•cosB + cosA•sinB

✓ sin(A-B) = sinA•cosB - cosA•sinB

✓ cos(A+B) = cosA•cosB - sinA•sinB

✓ cos(A-B) = cosA•cosB + sinA•sinB

Given:

sin@ + cos@ = √3

To prove:

tan@ + cot@ = 1

Proof:

We have;

sin@ + cos@ = √3 --------(1)

Squaring both sides of eq-(1) ,

We have ;

=> (sin@ + cos@)^2 = (√3)^2

=> (sin@)^2+(cos@)^2+2•sin@•cos@=3

{ (sin@)^2 + (cos@)^2 = 1 }

=> 1 + 2•sin@•cos@ = 3

=> 2•sin@•cos@ = 3 - 1

=> 2•sin@•cos@ = 2

=> sin@•cos@ = 2/2

=> sin@•cos@ = 1 ----------(2)

Now,

We have LHS,

= tan@ + cot@

= sin@/cos@ + cos@/sin@

= { (sin@)^2 + (cos@)^2 }/cos@•sin@

{ (sin@)^2 + (cos@)^2 = 1 }

= 1/sin@•cos@ { using eq-(2) }

= 1/1

= 1

= RHS

Hence proved.