Question

1. Prove the following identities ​

Answer 1

Explanation:

`\sf 1.1)(Sin \ \theta-Cos \ \theta)/(Sin \ \theta + Cos \ \theta)=((Sin \ \theta-Cos \ \theta))/(((Sin \ \theta + Cos \ \theta))*((Sin \ \theta-Cos \ \theta))/((Sin \ \theta-Cos \ \theta))`

`\sf = ((Sin \ \theta-Cos \ \theta)^2)/(Sin^2 \ \theta-Cos^2 \ \theta)\\\\=(Sin^2 \ \theta+Cos^2 \ \theta-2Sin \ \theta \ Cos\ \theta)/((Sin \ \theta-Cos \ \theta))\\\\\\\bf Identity: \ (a +b)^2= a^2 + b^2 - 2ab\\\\\\=(1-2Sin \ \theta \ Cos \ \theta)/((Sin \ \theta-Cos \ \theta)) = LHS`

`\sf 1.2) LHS = tan^2 \ x - Sin^2 \ x = (Sin^2 \ x)/(Cos^2 \ x)-Sin^2 \ x`

`\sf =(Sin^2 \ x)/(Cos^2 \ x)-(Sin^2 \ x*Cos^2 \ x)/(1*Cos^2 \ x)\\\\\\ = (Sin^2 \ x - Sin^2 \ x*Cos^2 \ x)/(Cos^2 \ x)\\\\\\= (Sin^2 \ x *(1 -Cos^2 \ x))/(Cos^2 \ x)\\\\=(Sin^2 \ x*Sin^2 \ x)/(Cos^2 \ x) \\\\ \bf 1 - Cos^2 \ x = Sin^2 \ x\\\\= (Sin^2 \ x)/(Cos^2 \ x)*Sin^2 \ x\\\\=tan^2 \ x * Sin^2 \ x = RHS`

`\sf 1.3) LHS = (1-Cos \ x)/(Sin \ x)-(Sin \ x)/(1+Cos \ x) =((1-Cos \ x)(1+Cos \ x))/(Sin \ x*(1+Cos \ x))-(Sin \ x*Sin \ x)/((1+Cos \ x)*Sin \ x)\\`

`\sf =(1 - Cos^2 \ x)/(Sin \ x*(1+Cos \ x))-(Sin^2 \ x)/(Sin \ x*(1+Cos \ x))\\\\=(Sin^2 \ x)/(Sin \ x*(1+Cos \ x)) - (Sin^2 \ x)/(Sin \ x*(1+Cos \ x))\\\\=(Sin^2 x - Sin^2 \ x)/(Sin \ x*(1+Cos \ x)) \\\\= 0 = RHS`

`\sf 1.4) LHS = Sin x - (1)/(Sin \ x + Cos \ x)+Cos \ x \\`

`\sf = (Sin \ x *(Sin \ x + Cos \ x) - 1 + Cos \ x * (Sin \ x + Cos \ x))/(Sin \ x + Cos \ x )\\\\\\= (Sin \ x * Sin \ x + Sin \ x*Cos \ x -1 + Cos \ x*Sin \ x + Cos \ x*Cos \ x)/(Sin \ x + Cos \ x)\\\\\\=(Sin^2 \x + Sin \ x \ Cos \ x - 1 + Cos \ x \ Sin \ x + Cos^2 \x)/(Sin \ x + Cos \ x)\\\\=(Sin^2 \ x + Cos^2 \ x - 1 + Sin \ xCos \x +Sin \ x Cos \ x)/(Sin \ x + Cos \ x)\\\\= (1 - 1 +2Sin \ x Cos \ x)/(Sin \ x + Cos \ x)\\\\= (2Sin \ x Cos \ x)/(Sin \ x + Cos \ x)`