Question

hi friends gm

solve fastly

Answer 1

$\huge \textit{\underline{\underline{Proof...}}}$

$\large \textit{LHS...}$

$\displaystyle \Longrightarrow\ \ \frac{(\sin\theta+1)-\cos\theta}{\sin\theta-1+\cos\theta} \\ \\ \\ \Longrightarrow\ \ \frac{\sin\theta+1-\cos\theta}{\sin\theta-1+\cos\theta}$

$\displaystyle \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)\frac{1+\sin\theta}{\cos^2\theta}}{(\sin\theta-1+\cos\theta)\frac{\sin\theta+1}{\cos^2\theta}} \\ \\ \\ \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)(1+\sin\theta)\frac{1}{\cos^2\theta}}{(\sin\theta-1+\cos\theta)(\sin\theta+1)\frac{1}{\cos^2\theta}}$

$\displaystyle \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)(\sin\theta+1)\frac{1}{\cos^2\theta}}{(\sin\theta-1)(\sin\theta+1)+\cos\theta(1+\sin\theta)\frac{1}{\cos^2\theta}} \\ \\ \\ \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)(\sin\theta+1)\frac{1}{\cos^2\theta}}{\sin^2\theta-1+\cos\theta(1+\sin\theta)\frac{1}{\cos^2\theta}} \\ \\ \\ \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)(\sin\theta+1)\frac{1}{\cos^2\theta}}{-\cos^2\theta+\cos\theta(1+\sin\theta)\frac{1}{\cos^2\theta}}$

$\displaystyle \Longrightarrow\ \ \frac{(\sin\theta+1-\cos\theta)(\sin\theta+1)\frac{1}{\cos^2\theta}}{\cos\theta(\sin\theta+1-\cos\theta)\frac{1}{\cos^2\theta}} \\ \\ \\ \Longrightarrow\ \ \frac{(\sin\theta+1)\frac{1}{\cos^2\theta}}{\frac{1}{\cos\theta}} \\ \\ \\ \Longrightarrow\ \ \frac{\sin\theta+1}{\cos^2\theta} \ \div \ \frac{1}{\cos\theta} \\ \\ \\ \Longrightarrow\ \ \frac{1+\sin\theta}{\cos^2\theta} \times \cos\theta$

$\displaystyle \Longrightarrow\ \ \frac{1+\sin\theta}{\cos\theta}$

$\displaystyle \Longrightarrow\ \ \large \textit{...RHS}$

$\huge \textsc{\underline{\underline{Hence Proved!!!}}}$

Answer 2

Solution:-

Note:-

Taking A in place of Theta!!

Proof:-

Dividing Numerator & Denominator by CosA.

=> (TanA + SecA - 1)/ (TanA - SecA + 1)

=> [TanA + SecA - ( Sec²A - Tan²A)]/ (TanA - SecA +1)

=> [ TanA + SecA - (SecA - TanA)(SecA + TanA)]/(TanA - SecA +1)

Taking (TanA + SecA) Common in Numerator.

=> (TanA + SecA) ( 1-SecA + TanA)/(1-SecA+TanA)

=> TanA + SecA

=> SinA/CosA + 1/CosA

=> (SinA+1)/CosA

Hence Proved!!