Prove the trigonometric identity $\frac{\left(\cos\left(x\right)+\sin\left(x\right)\right)^2}{\sec\left(x\right)^2+2\tan\left(x\right)}=\cos\left(x\right)^2$

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$\frac{\left(\cos\left(x\right)+\sin\left(x\right)\right)^2}{\sec\left(x\right)^2+2\tan\left(x\right)}$

Learn how to solve proving trigonometric identities problems step by step online. Prove the trigonometric identity ((cos(x)+sin(x))^2)/(sec(x)^2+2tan(x))=cos(x)^2. Starting from the left-hand side (LHS) of the identity. Rewrite \sec\left(x\right)^2+2\tan\left(x\right) in terms of sine and cosine functions. Divide fractions \frac{\left(\cos\left(x\right)+\sin\left(x\right)\right)^2}{\frac{1+2\sin\left(x\right)\cos\left(x\right)}{\cos\left(x\right)^2}} with Keep, Change, Flip: a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}. Using the sine double-angle identity: \sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right).