Question

prove that 1 upon sec theta minus tan theta minus one upon cos theta is equal to 1 upon cos theta minus 1 upon sec theta + tan theta​

Answer 1

Answer:

1/(SecФ - TanФ) - 1/CosФ = 1/CosФ - 1/(SecФ + TanФ).... Proved

Step-by-step explanation:

We have to prove that,

1/(SecФ - TanФ) - 1/CosФ = 1/CosФ - 1/(SecФ + TanФ) .... (1)

Let us rearrange the LHS and RHS and we will get

1/(SecФ - TanФ)+1/(SecФ + TanФ)=1/CosФ+1/CosФ  .... (2)

So we can prove equation (2) then equation (1) will automatically be proved.

Now, LHS of equation (2)

= 1/(SecФ - TanФ)+1/(SecФ + TanФ)

= (SecФ - TanФ+SecФ + TanФ)/ {(SecФ - TanФ)×(SecФ + TanФ)}

= 2SecФ/(Sec²Ф-Tan²Ф)

=2 SecФ { As (Sec²Ф-Tan²Ф)=1}

=2/CosФ

=1/CosФ+1/CosФ

=RHS of equation (2).... Proved

So, as the equation (2) is proved, then we can conclude that equation (1) is also proved.