Question

If sin 0 + cos O =√3 then prove that tan 0 + cot 0 = 1​

Answer 1

Question:

If sin@ + cos@ = √3 , then prove that,

tan@ + cot@ = 1.

Note:

• (A+B)^2 = A^2 + B^2 + 2•A•B

• (sin@)^2 + (cos@)^2 = 1

• (tan@)^2 + 1 = (sec@)^2

• (cot@)^2 + 1 = (cosec@)^2

• tan@ = sin@/cos@

• cot@ = 1/tan@ = cos@/sin@

Solution:

Given:

sin@ + cos@ = √3

To prove:

tan@ + cot@ = 1

Proof:

We have;

sin@ + cos@ = √3 --------(1)

Now,

Squaring both sides of eq-(1) ,

We get;

=> (sin@ + cos@)^2 = (√3)^2

=> (sin@)^2+(cos@)^2+2•sin@•cos@=3

=> 1 + 2•sin@•cos@ = 3

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

=> 2•sin@•cos@ = 3 - 1

=> 2•sin@•cos@ = 2

=> sin@•cos@ = 2/2

=> sin@•cos@ = 1

Now,

We have LHS

= tan@ + cot@

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

={(sin@)^2 + (cos@)^2}/cos@•sin@

= 1/sin@•cos@

= 1/1

= 1

= RHS

Hence proved.